Statistics and Probability

Statistics, Frequency and Frequency Distributions

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Statistics: - By “statistics” we mean aggregate or combination of facts affected to a marked extends by multiplicity of causes, numerically expressed and estimated according to reasonable standards of accuracy, collected in a systematic manner and placed in relation to each other.

There are two ways of statistics:

1. Frequency Distribution and

2. Graphical Distribution.

Frequency: - The way to count the number of items a particular value is repeated, is called the frequency of any class.

That means, frequency is the total number of items that a particular value is repeated in a table or data.

Frequency Distributions: - A set of classes together with the frequencies of occurrence of values in again set of data, presented in a tabular form, is referred to as a frequency distribution.

Construct a frequency distribution from the class marks of EEE 36 Batch who got the numbers in total trimester in statistics and probability.

Course:

97, 13, 81, 25, 37, 55, 19, 33, 59, 46, 67, 43, 12, 87, 90, 65, 76, 81, 79, 13, 5, 12, 35, 17, 46, 65, 43, 12, 85, 93,

Solution:

Here, the lowest value=5

And the highest value=97

For this kind of ungrouped data we have to choose k in such a way that 2k ≥ number of variables.

Here, k=5

So 2k = 25 = 32 > 30

Empirical Relation between Mean, Median and Mode

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A distribution in which the values of mean, median and mode coincide (i.e. mean = median = mode) is known as a symmetrical distribution. Conversely, when values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed distribution. In moderately skewed or asymmetrical distribution a very important relationship exists among these three measures of central tendency. In such distributions the distance between the mean and median is about one-third of the distance between the mean and mode, as will be clear from the diagrams 1 and 2 Karl Pearson expressed this relationship as:

Measures of Central Tendency

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Central Tendency: - The tendency of the individual item of a statistical series to cluster around the central value is called the Central Tendency. Sometimes it is called the measure of location or a measure of representation.

Several types of Central Tendency can be defined: The commons are

The Arithmetic Mean
The Median
The Mode
The Geometric Mean
The Harmonic Mean

The Arithmetic Mean: - The Arithmetic Mean of a grouped frequency distribution is defined as

A = any guessed or assumed class mark.

f = Frequency of each class interval.

n = Sum of total frequency.

i = Range of class interval.

d = Deviation of the assumed class mark from each class interval by the range of class interval.

d = (Xi – A) / i

The Median: - The Median of a grouped is defined as

Where,

M_e = Median of the total class.

f_c = Previous cumulative frequency of all classes above the media class.

f_m = Frequency of the corresponding class interval.

i = range of class interval.

L = Lower class boundary of median class.

n = Sum of total frequency.

The Mode: - The Mode of a set of number is that value which occurs with the greatest frequency.

The Mode for a grouped data/frequency distribution is denoted by

Where,

L = Lower limit of modal class interval.

∆₁ = Difference between modal and pre-modal group.

∆₂ = Difference between modal and post-modal group

i = range of class interval.

The Geometric Mean: - The Geometric Mean is for a grouped frequency distribution is denoted by

Where,

G = Geometric Mean

n = sum of total frequency.

f_i = Frequency corresponding each class interval.

x_i = Class Mark.

The Harmonic Mean: - The Harmonic Mean H for a grouped frequency distribution is

Where,

H = Harmonic Mean.

n = sum of total frequency.

f_i = Frequency corresponding each class interval.

x_i = Class Mark.

Problem: -

The given frequency is the efficiency score of 115 students in their 70% marks. Find the Arithmetic Mean, Median, Mode, Geometric Mean and Harmonic Mean.

Solution:

Some special measurements following any section of Central Tendency:

Quartiles
Deciles and
Percentile

Quartiles: - The Quartiles are those values in a series which divide the total frequency into four equal parts. It is denoted by Q where

Where,

r = 1, 2, 3,…….

L_r = Lower limit of the Quartiles class,

n = Sum of the total frequency,

r = Position of Quartiles,

F_r = Cumulative frequency of the pre-rth Quartiles class,

f_r = Corresponding frequency,

i = Range of class interval.

Deciles: - The Deciles are those values in a series which divide the total frequency into ten equal parts. It is denoted by D where

Where,

r = 1, 2, 3,…….

L_r = Lower limit of the Deciles class,

n = Sum of the total frequency,

r = Position of Deciles,

F_r = Cumulative frequency of the pre-rth Deciles class,

f_r = Corresponding frequency,

i = Range of class interval.

Percentiles: - The Percentiles are those values in a series which divide the total frequency into 100 equal parts. It is denoted by P where

Where,

r = 1, 2, 3,…….

L_r = Lower limit of the Percentiles class,

n = Sum of the total frequency,

r = Position of Percentiles,

F_r = Cumulative frequency of the pre-rth Percentiles class,

f_r = Corresponding frequency,

i = Range of class interval.

Graphical Distribution of Frequency Distribution

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Frequency distribution can be presented graphically in any one of the following ways:

Histogram
Frequency Polygon
Smooth Frequency Curve
Cumulative Frequency Curve of Ogive Curve
Pie-Chart

Histogram: - A histogram is an area diagram in which the frequencies corresponding to each class interval of frequency distribution are by the area of a rectangle without leaving no gap between the cosective rectangles.

Frequency Polygon: - This is one kind of histogram which is represented by joining the straight lines of the mid points of the upper horizontal side of each rectangle with adjacent rectangles.

Smooth Frequency Curve: - This is one kind of histogram which is represented by joining the mid points by free hand of the upper horizontal side of each rectangle with adjacent rectangles.

Comulative Frequency Curve or Ogive Curve: - The total frequency of all values less then the upper class boundary of a given class interval is called the Cumulative Frequency up to and including that class interval. The graph of such a distribution is called a Cumulative Frequency or Ogive.

There are two methods of constructing Ogive, namely

The less than method and
The more than method

The less than method: - In the “less than” method, we start with the upper boundary of each class interval and cumulative frequencies; when the frequencies are plotted, we get a rising curve.

The more than method: - In the “more than” method, we start with the lower boundary-rise of each class interval and from the total frequencies we substrate the frequency of each class when these frequencies are plotted we get a declining curve.

Draw a 'less than' ogive curve for the following data:

To Plot an Ogive:

(i) We plot the points with coordinates having abscissae as actual limits and ordinates as the cumulative frequencies, (10, 2), (20, 10), (30, 22), (40, 40), (50, 68), (60, 90), (70, 96) and (80, 100) are the coordinates of the points.

(ii) Join the points plotted by a smooth curve.

(iii) An Ogive is connected to a point on the X-axis representing the actual lower limit of the first class.

Scale:

X -axis 1 cm = 10 marks, Y -axis 1cm = 10 c.f.

Using the data given below, construct a 'more than' cumulative frequency table and draw the Ogive.

To Plot an Ogive

(i) We plot the points with coordinates having abscissae as actual lower limits and ordinates as the cumulative frequencies,

(70.5, 2), (60.5, 7), (50.5, 13), (40.5, 23), (30.5, 37), (20.5, 49),

(10.5, 57), (0.5, 60) are the coordinates of the points.

(ii) Join the points by a smooth curve.

(iii) An Ogive is connected to a point on the X-axis representing the actual upper limit of the last class [in this case) i.e., point (80.5, 0)].

Scale:

X-axis 1 cm = 10 marks

Y-axis 2 cm = 10 c.f

To reconstruct frequency distribution from cumulative frequency distribution.

Pie-Chart: -Represent the following distribution by Pie-Chart: -

Problem: -

Distribution of 100 students classified according to the marks that they securated in a class for an equal class interval.

Class Intervals	Frequency
60-65	8
65-70	20
70-75	27
75-80	15
80-90	10
90-100	20

Measure of Dispersion

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Dispersion: - Dispersion refers to the scatteredness of the individual items of statistical series from their central value. So a descriptive measure of scatter of the values about the average is called measure of Dispersion.

The followings are the important methods for measure of dispersion:

The Range
The average/mean Deviation
Quartile Deviation
The 10 – 90 Percentile Range
The Standard Deviation
The Variance

The Range: - The Range of a set of numbers is the difference between the largest and smallest numbers in the set.

The average/mean Deviation: - The average/mean Deviation, of a set of N numbers X₁, X₂,……..X_N is abbreviated MD and is defined by

Quartile Deviation: - Quartile Deviation, of a set of data is denoted by Q and defined by,Q = (Q ₃ – Q₁)/2

The 10 – 90 Percentile Range: - The 10 – 90 Percentile Range, of a set of data is defined by, 10 – 90 Percentile Range = P₉₀ – P₁₀

The Standard Deviation: - The Standard Deviation of a set of N numbers X₁, X₂,……..X_N is denoted by σ and is defined by

The Variance: - The Variance of a set of data is defined as the square of the standard deviation and is thus given by σ² and is denoted by

Co-efficient of Variation: - If the average of a statistical data is the mean x and if the absolute dispersion is the standard deviation, then the relative dispersion is called Co-efficient of dispersion. It is denoted by v and is given by, v = (σ/x) × 100

Problem: -

Find the standard deviation and co-efficient of variation of the class test of 100 students of XYZ University.

Moments

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Moments: - Moments are certain mathematical constants used to as certain the nature and form of a frequency distribution. Moments in statistics are used to describe the various characteristic of a frequency distribution like Central Tendency, Dispersion, Skewness and Kurtosis. It is symbolized by the Greek letter μ

There are two Moments:

Raw Moments and
Corrected Moments

Raw Moments for grouped data,

Relation between Raw Moments and Corrected Moments for grouped data:

Skewness

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Skewness is a measure of the degree of asymmetry of a distribution. If the left tail (tail at small end of the distribution) is more pronounced than the right tail (tail at the large end of the distribution), the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness.

Several types of skewness are defined, the terminology and notation of which are unfortunately rather confusing. "The" skewness of a distribution is defined to be

(1)

Positively Skewed Distribution: - The value of the arithmetic mean is greater than the mode; then the distribution is called Positively Skewed.

Nagatively Skewed Distribution: - If the value of the mode is greater than the arithmetic mean; the distribution is called Negatively Skewed.

Several forms of skewness are also defined. The momental skewness is defined by

(2)

The Pearson mode skewness is defined by

(3)

Pearson's skewness coefficients are defined by

(4)

and

(5)

The Bowley skewness (also known as quartile skewness coefficient) is defined by

(6)

where the s denote the interquartile ranges. The momental skewness is

Kurtosis

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Kurtosis is the degree of peakedness of a distribution, defined as a normalized form of the fourth central moment of a distribution. There are several flavors of kurtosis commonly encountered, including the kurtosis proper, denoted or defined by

(1)

where denotes the th central moment (and in particular, is the variance). This form is implemented in Mathematica as Kurtosis[dist].

The kurtosis "excess" is denoted or , is defined by

(2)

and is implemented in Mathematica as KurtosisExcess[dist]. Kurtosis excess is commonly used because of a normal distribution is equal to 0, while the kurtosis proper is equal to 3.

Unfortunately, Abramowitz and Stegun (1972) confusingly refer to as the "excess or kurtosis."

Lepto-Kurtic: - If a curve is more peaked than normal curve then it is colled Lepto-Kurtic.

Platy-Kurtic: - If a curve is more flat-tapped than normal curve then it is called Platy-Kurtic.

Meso-Kurtic: -The curve representing a normal shape in a frequency distribution is called Meso-Kurtic.

Correlation and Linearity

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Correlation coefficients measure the strength of association between two variables. The most common correlation coefficient, called the Pearson product-moment correlation coefficient, measures the strength of the linear association between variables.

In this tutorial, when we speak simply of a correlation coefficient, we are referring to the Pearson product-moment correlation. Generally, the correlation coefficient of a sample is denoted by r, and the correlation coefficient of a population is denoted by ρ or R.

How to Interpret a Correlation Coefficient

The sign and the absolute value of a correlation coefficient describe the direction and the magnitude of the relationship between two variables.

The value of a correlation coefficient ranges between -1 and 1.
The greater the absolute value of a correlation coefficient, the stronger the linear relationship.
The strongest linear relationship is indicated by a correlation coefficient of -1 or 1.
The weakest linear relationship is indicated by a correlation coefficient equal to 0.
A positive correlation means that if one variable gets bigger, the other variable tends to get bigger.
A negative correlation means that if one variable gets bigger, the other variable tends to get smaller.

Keep in mind that the Pearson product-moment correlation coefficient only measures linear relationships. Therefore, a correlation of 0 does not mean zero relationship between two variables; rather, it means zero linear relationship. (It is possible for two variables to have zero linear relationship and a strong curvilinear relationship at the same time.)

Scatterplots and Correlation Coefficients

The scatterplots below show how different patterns of data produce different degrees of correlation.


Maximum positive correlation (r = 1.0)	Strong positive correlation (r = 0.80)

Zero correlation
(r = 0)


Minimum negative correlation (r = -1.0)	Moderate negative correlation (r = -0.43)

Strong correlation & outlier
(r = 0.71)

Several points are evident from the scatterplots.

When the slope of the line in the plot is negative, the correlation is negative; and vice versa.
The strongest correlations (r = 1.0 and r = -1.0 ) occur when data points fall exactly on a straight line.
The correlation becomes weaker as the data points become more scattered.
If the data points fall in a random pattern, the correlation is equal to zero.
Correlation is affected by outliers. Compare the first scatterplot with the last scatterplot. The single outlier in the last plot greatly reduces the correlation (from 1.00 to 0.71).

How to Calculate a Correlation Coefficient

If you look in different statistics textbooks, you are likely to find different-looking (but equivalent) formulas for computing a correlation coefficient. In this section, we present several formulas that you may encounter.

The most common formula for computing a product-moment correlation coefficient (r) is given below.

Product-moment correlation coefficient. The correlation r between two variables is:

r = Σ (xy) / sqrt [ ( Σ x² ) * ( Σ y² ) ]

where Σ is the summation symbol, x = x_i - x, x_i is the x value for observation i, x is the mean x value, y = y_i - y, y_i is the y value for observation i, and y is the mean y value.

The formula below uses population means and population standard deviations to compute a population correlation coefficient (ρ) from population data.

Population correlation coefficient. The correlation ρ between two variables is:

ρ = [ 1 / N ] * Σ { [ (X_i - μ_X) / σ_x ] * [ (Y_i - μ_Y) / σ_y ] }

Where N is the number of observations in the population, Σ is the summation symbol, X_i is the X value for observation i, μ_X is the population mean for variable X, Y_i is the Y value for observation i, μ_Y is the population mean for variable Y, σ_x is the population standard deviation of X, and σ_y is the population standard deviation of Y.

The formula below uses sample means and sample standard deviations to compute a correlation coefficient (r) from sample data.

Sample correlation coefficient. The correlation r between two variables is:

r = [ 1 / (n - 1) ] * Σ { [ (x_i - x) / s_x ] * [ (y_i - y) / s_y ] }

where n is the number of observations in the sample, Σ is the summation symbol, x_i is the x value for observation i, x is the sample mean of x, y_i is the y value for observation i, y is the sample mean of y, s_x is the sample standard deviation of x, and s_y is the sample standard deviation of y.

The interpretation of the sample correlation coefficient depends on how the sample data is collected. With a simple random sample, the sample correlation coefficient is an unbiased estimate of the population correlation coefficient.

Each of the latter two formulas can be derived from the first formula. Use the second formula when you have data from the entire population. Use the third formula when you only have sample data. When in doubt, use the first formula. It is always correct.

Fortunately, you will rarely have to compute a correlation coefficient by hand. Many software packages (e.g., Excel) and most graphing calculators have a correlation function that will do the job for you.

Note: Sometimes, it is not clear whether a software package or a graphing calculator is computing a population correlation coefficient or a sample correlation coefficient. For example, a casual user might not realize that Microsoft uses a population correlation coefficient (ρ) for the Pearson function in its Excel software.

Problem 1

A national consumer magazine reported the following correlations.

The correlation between car weight and car reliability is -0.30.
The correlation between car weight and annual maintenance cost is 0.20.

Which of the following statements are true?

I. Heavier cars tend to be less reliable.
II. Heavier cars tend to cost more to maintain.
III. Car weight is related more strongly to reliability than to maintenance cost.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

Solution

The correct answer is (E). The correlation between car weight and reliability is negative. This means that reliability tends to decrease as car weight increases. The correlation between car weight and maintenance cost is positive. This means that maintenance costs tend to increase as car weight increases.

The strength of a relationship between two variables is indicated by the absolute value of the correlation coefficient. The correlation between car weight and reliability has an absolute value of 0.30. The correlation between car weight and maintenance cost has an absolute value of 0.20. Therefore, the relationship between car weight and reliability is stronger than the relationship between car weight and maintenance cost.

Basic Probability

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The probability of a sample point is a measure of the likelihood that the sample point will occur.

Probability of a Sample Point

By convention, statisticians have agreed on the following rules.

The probability of any sample point can range from 0 to 1.
The sum of probabilities of all sample points in a sample space is equal to 1.

Example 1
Suppose we conduct a simple statistical experiment. We flip a coin one time. The coin flip can have one of two outcomes - heads or tails. Together, these outcomes represent the sample space of our experiment. Individually, each outcome represents a sample point in the sample space. What is the probability of each sample point?

Solution: The sum of probabilities of all the sample points must equal 1. And the probability of getting a head is equal to the probability of getting a tail. Therefore, the probability of each sample point (heads or tails) must be equal to 1/2.

Example 2
Let's repeat the experiment of Example 1, with a die instead of a coin. If we toss a fair die, what is the probability of each sample point?

Solution: For this experiment, the sample space consists of six sample points: {1, 2, 3, 4, 5, 6}. Each sample point has equal probability. And the sum of probabilities of all the sample points must equal 1. Therefore, the probability of each sample point must be equal to 1/6.

Probability of an Event

The probability of an event is a measure of the likelihood that the event will occur. By convention, statisticians have agreed on the following rules.

The probability of any event can range from 0 to 1.
The probability of event A is the sum of the probabilities of all the sample points in event A.
The probability of event A is denoted by P(A).

Thus, if event A were very unlikely to occur, then P(A) would be close to 0. And if event A were very likely to occur, then P(A) would be close to 1.

Example 1
Suppose we draw a card from a deck of playing cards. What is the probability that we draw a spade?

Solution: The sample space of this experiment consists of 52 cards, and the probability of each sample point is 1/52. Since there are 13 spades in the deck, the probability of drawing a spade is

P(Spade) = (13)(1/52) = 1/4

Example 2
Suppose a coin is flipped 3 times. What is the probability of getting two tails and one head?

Solution: For this experiment, the sample space consists of 8 sample points.

S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH}

Each sample point is equally likely to occur, so the probability of getting any particular sample point is 1/8. The event "getting two tails and one head" consists of the following subset of the sample space.

A = {TTH, THT, HTT}

The probability of Event A is the sum of the probabilities of the sample points in A. Therefore,

P(A) = 1/8 + 1/8 + 1/8 = 3/8

Probability Distributions

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To understand probability distributions, it is important to understand variables. random variables, and some notation.

A variable is a symbol (A, B, x, y, etc.) that can take on any of a specified set of values.
When the value of a variable is the outcome of a statistical experiment, that variable is a random variable.

Generally, statisticians use a capital letter to represent a random variable and a lower-case letter, to represent one of its values. For example,

X represents the random variable X.
P(X) represents the probability of X.
P(X = x) refers to the probability that the random variable X is equal to a particular value, denoted by x. As an example, P(X = 1) refers to the probability that the random variable X is equal to 1.

Probability Distributions

An example will make clear the relationship between random variables and probability distributions. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of Heads that result from this experiment. The variable X can take on the values 0, 1, or 2. In this example, X is a random variable; because its value is determined by the outcome of a statistical experiment.

A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurence. Consider the coin flip experiment described above. The table below, which associates each outcome with its probability, is an example of a probability distribution.

Number of heads	Probability
0	0.25
1	0.50
2	0.25

The above table represents the probability distribution of the random variable X.

Cumulative Probability Distributions

A cumulative probability refers to the probability that the value of a random variable falls within a specified range.

Let us return to the coin flip experiment. If we flip a coin two times, we might ask: What is the probability that the coin flips would result in one or fewer heads? The answer would be a cumulative probability. It would be the probability that the coin flip experiment results in zero heads plus the probability that the experiment results in one head.

P(X < 1) = P(X = 0) + P(X = 1) = 0.25 + 0.50 = 0.75

Like a probability distribution, a cumulative probability distribution can be represented by a table or an equation. In the table below, the cumulative probability refers to the probability than the random variable X is less than or equal to x.

Number of heads: x	Probability: P(X = x)	Cumulative Probability: P(X < x)
0	0.25	0.25
1	0.50	0.75
2	0.25	1.00

Uniform Probability Distribution

The simplest probability distribution occurs when all of the values of a random variable occur with equal probability. This probability distribution is called the uniform distribution.

Uniform Distribution. Suppose the random variable X can assume k different values. Suppose also that the P(X = x_k) is constant. Then,
P(X = x_k) = 1/k

Example 1

Suppose a die is tossed. What is the probability that the die will land on 6 ?

Solution: When a die is tossed, there are 6 possible outcomes represented by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is a random variable (X), and each outcome is equally likely to occur. Thus, we have a uniform distribution. Therefore, the P(X = 6) = 1/6.

Example 2

Suppose we repeat the dice tossing experiment described in Example 1. This time, we ask what is the probability that the die will land on a number that is smaller than 5 ?

Solution: When a die is tossed, there are 6 possible outcomes represented by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is equally likely to occur. Thus, we have a uniform distribution.

This problem involves a cumulative probability. The probability that the die will land on a number smaller than 5 is equal to:

P( X < x =" 1)" x =" 2)" x =" 3)" x =" 4)" 6 =" 2/3

Rules of Probability

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Often, we want to compute the probability of an event from the known probabilities of other events. This lesson covers some important rules that simplify those computations.

Definitions and Notation

Before discussing the rules of probability, we state the following definitions:

Two events are mutually exclusive or disjoint if they cannot occur at the same time.
The probability that Event A occurs, given that Event B has occurred, is called a conditional probability. The conditional probability of Event A, given Event B, is denoted by the symbol P(A|B).
The complement of an event is the event not occuring. The probability that Event A will not occur is denoted by P(A').
The probability that Events A and B both occur is the probability of the intersection of A and B. The probability of the intersection of Events A and B is denoted by P(A ∩ B). If Events A and B are mutually exclusive, P(A ∩ B) = 0.
The probability that Events A or B occur is the probability of the union of A and B. The probability of the union of Events A and B is denoted by P(A ∪ B) .
If the occurence of Event A changes the probability of Event B, then Events A and B are dependent. On the other hand, if the occurence of Event A does not change the probability of Event B, then Events A and B are independent.

Rule of Subtraction

In a previous lesson, we learned two important properties of probability:

The probability of an event ranges from 0 to 1.
The sum of probabilities of all possible events equals 1.

The rule of subtraction follows directly from these properties.

Rule of Subtraction The probability that event A will occur is equal to 1 minus the probability that event A will not occur.

P(A) = 1 - P(A')

Suppose, for example, the probability that Bill will graduate from college is 0.80. What is the probability that Bill will not graduate from college? Based on the rule of subtraction, the probability that Bill will not graduate is 1.00 - 0.80 or 0.20.

Rule of Multiplication

The rule of multiplication applies to the situation when we want to know the probability of the intersection of two events; that is, we want to know the probability that two events (Event A and Event B) both occur.

Rule of Multiplication The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred.

P(A ∩ B) = P(A) P(B|A)

Example
An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn without replacement from the urn. What is the probability that both of the marbles are black?

Solution: Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, there are 9 marbles in the urn, 3 of which are black. Therefore, P(B|A) = 3/9.

Therefore, based on the rule of multiplication:

P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(3/9) = 12/90 = 2/15

Rule of Addition

The rule of addition applies to the following situation. We have two events, and we want to know the probability that either event occurs.

Rule of Addition The probability that Event A and/or Event B occur is equal to the probability that Event A occurs plus the probability that Event B occurs minus the probability that both Events A and B occur.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B))

Note: Invoking the fact that P(A ∩ B) = P( A )P( B | A ), the Addition Rule can also be expressed as

P(A ∪ B) = P(A) + P(B) - P(A)P( B | A )

Example
A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.30, , and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?

Solution: Let F = the event that the student checks out fiction; and let N = the event that the student checks out non-fiction. Then, based on the rule of addition:

P(F ∪ N) = P(F) + P(N) - P(F ∩ N)
P(F ∪ N) = 0.40 + 0.30 - 0.20 = 0.50

Problem 1

An urn contains 6 red marbles and 4 black marbles. Two marbles are drawn with replacement from the urn. What is the probability that both of the marbles are black?

(A) 0.16
(B) 0.32
(C) 0.36
(D) 0.40
(E) 0.60

Solution

The correct answer is A. Let A = the event that the first marble is black; and let B = the event that the second marble is black. We know the following:

In the beginning, there are 10 marbles in the urn, 4 of which are black. Therefore, P(A) = 4/10.
After the first selection, we replace the selected marble; so there are still 10 marbles in the urn, 4 of which are black. Therefore, P(B|A) = 4/10.

Therefore, based on the rule of multiplication:

P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10)*(4/10) = 16/100 = 0.16

Problem 2

A card is drawn randomly from a deck of ordinary playing cards. You win $10 if the card is a spade or an ace. What is the probability that you will win the game?

(A) 1/13
(B) 13/52
(C) 4/13
(D) 17/52
(E) None of the above.

Solution

The correct answer is C. Let S = the event that the card is a spade; and let A = the event that the card is an ace. We know the following:

There are 52 cards in the deck.
There are 13 spades, so P(S) = 13/52.
There are 4 aces, so P(A) = 4/52.
There is 1 ace that is also a spade, so P(S ∩ A) = 1/52.

Therefore, based on the rule of addition:

P(S ∪ A) = P(S) + P(A) - P(S ∩ A)
P(S ∪ A) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13

Random Variables

2010-01-29T07:26:00.000-08:00

When the numerical value of a variable is determined by a chance event, that variable is called a random variable.

Discrete vs. Continuous Random Variables

Random variables can be discrete or continuous.

Discrete. Discrete random variables take on integer values, usually the result of counting. Suppose, for example, that we flip a coin and count the number of heads. The number of heads results from a random process - flipping a coin. And the number of heads is represented by an integer value - a number between 0 and plus infinity. Therefore, the number of heads is a discrete random variable.
Continuous. Continuous random variables, in contrast, can take on any value within a range of values. For example, suppose we flip a coin many times and compute the average number of heads per flip. The average number of heads per flip results from a random process - flipping a coin. And the average number of heads per flip can take on any value between 0 and 1, even a non-integer value. Therefore, the average number of heads per flip is a continuous random variable.

Problem 1

Which of the following is a discrete random variable?

I. The average height of a randomly selected group of boys.
II. The annual number of sweepstakes winners from New York City.
III. The number of presidential elections in the 20th century.

(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III

Solution

The correct answer is B. The annual number of sweepstakes winners is an integer value and it results from a random process; so it is a discrete random variable. The average height of a group of boys could be a non-integer, so it is not a discrete variable. And the number of presidential elections in the 20th century is an integer, but it does not vary and it does not result from a random process; so it is not a random variable.

Poisson Distribution

2010-01-29T06:56:00.000-08:00

A Poisson experiment is a statistical experiment that has the following properties:

The experiment results in outcomes that can be classified as successes or failures.
The average number of successes (μ) that occurs in a specified region is known.
The probability that a success will occur is proportional to the size of the region.
The probability that a success will occur in an extremely small region is virtually zero.

Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

Notation

The following notation is helpful, when we talk about the Poisson distribution.

e: A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.)
μ: The mean number of successes that occur in a specified region.
x: The actual number of successes that occur in a specified region.
P(x; μ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.

Poisson Distribution

A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution.

Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula:

Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e^-μ) (μ^x) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

The Poisson distribution has the following properties:

The mean of the distribution is equal to μ .
The variance is also equal to μ .

Example 1

The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?

Solution: This is a Poisson experiment in which we know the following:

μ = 2; since 2 homes are sold per day, on average.
x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.
e = 2.71828; since e is a constant equal to approximately 2.71828.

We plug these values into the Poisson formula as follows:

P(x; μ) = (e^-μ) (μ^x) / x!
P(3; 2) = (2.71828^-2) (2³) / 3!
P(3; 2) = (0.13534) (8) / 6
P(3; 2) = 0.180

Thus, the probability of selling 3 homes tomorrow is 0.180 .

Cumulative Poisson Probability

A cumulative Poisson probability refers to the probability that the Poisson random variable is greater than some specified lower limit and less than some specified upper limit.

Example 1

Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?

Solution: This is a Poisson experiment in which we know the following:

μ = 5; since 5 lions are seen per safari, on average.
x = 0, 1, 2, or 3; since we want to find the likelihood that tourists will see fewer than 4 lions; that is, we want the probability that they will see 0, 1, 2, or 3 lions.
e = 2.71828; since e is a constant equal to approximately 2.71828.

To solve this problem, we need to find the probability that tourists will see 0, 1, 2, or 3 lions. Thus, we need to calculate the sum of four probabilities: P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5). To compute this sum, we use the Poisson formula:

P(x < 3, 5) = P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5)
P(x < 3, 5) = [ (e^-5)(5⁰) / 0! ] + [ (e^-5)(5¹) / 1! ] + [ (e^-5)(5²) / 2! ] + [ (e^-5)(5³) / 3! ]
P(x < 3, 5) = [ (0.006738)(1) / 1 ] + [ (0.006738)(5) / 1 ] + [ (0.006738)(25) / 2 ] + [ (0.006738)(125) / 6 ]
P(x < 3, 5) = [ 0.0067 ] + [ 0.03369 ] + [ 0.084224 ] + [ 0.140375 ]
P(x < 3, 5) = 0.2650

Thus, the probability of seeing at no more than 3 lions is 0.2650.

Binomial Distribution

2010-01-28T06:08:00.000-08:00

The binomial distribution gives the discrete probability distribution of obtaining exactly successes out of Bernoulli trials (where the result of each Bernoulli trial is true with probability and false with probability ). The binomial distribution is therefore given by

			(1)
			(2)

where is a binomial coefficient. The above plot shows the distribution of successes out of trials with .

The binomial distribution is implemented in Mathematica as BinomialDistribution[n, p].

The probability of obtaining more successes than the observed in a binomial distribution is

(3)

where

(4)

is the beta function, and is the incomplete beta function.

The characteristic function for the binomial distribution is

(5)

(Papoulis 1984, p. 154). The moment-generating function for the distribution is

		" border="0" height="20" width="27">	(6)
			(7)
			(8)
			(9)
			(10)
			(11)

The mean is

			(12)
			(13)
			(14)

The moments about 0 are

			(15)
			(16)
			(17)
			(18)

so the moments about the mean are

			(19)
			(20)
			(21)

The skewness and kurtosis excess are

			(22)
			(23)
			(24)
			(25)

The first cumulant is

(26)

and subsequent cumulants are given by the recurrence relation

(27)

The mean deviation is given by

(28)

For the special case , this is equal to

			(29)
			(30)

where is a double factorial. For , 2, ..., the first few values are therefore 1/2, 1/2, 3/4, 3/4, 15/16, 15/16, ... (Sloane's A086116 and A086117). The general case is given by

(31)

Steinhaus (1999, pp. 25-28) considers the expected number of squares containing a given number of grains on board of size after random distribution of of grains,

(32)

Taking gives the results summarized in the following table.


0	23.3591
1	23.7299
2	11.8650
3	3.89221
4	0.942162
5	0.179459
6	0.0280109
7	0.0036840
8
9
10

An approximation to the binomial distribution for large can be obtained by expanding about the value where is a maximum, i.e., where . Since the logarithm function is monotonic, we can instead choose to expand the logarithm. Let , then

(33)

where

(34)

But we are expanding about the maximum, so, by definition,

(35)

This also means that is negative, so we can write . Now, taking the logarithm of (◇) gives

(36)

For large and we can use Stirling's approximation

(37)

			(38)
			(39)
			(40)
			(41)
			(42)

and

(43)

To find , set this expression to 0 and solve for ,

(44)

(45)

(46)

(47)

since . We can now find the terms in the expansion

			(48)
			(49)
			(50)
			(51)
			(52)
			(53)
			(54)
			(55)
			(56)
			(57)
			(58)
			(59)

Now, treating the distribution as continuous,

infty)sum_(n=0)^NP(n) approx intP(n)dn=int_(-infty)^inftyP(n^~+eta)deta=1. " border="0" height="48" width="285">

(60)

Since each term is of order smaller than the previous, we can ignore terms higher than , so

(61)

The probability must be normalized, so

(62)

and

			(63)
			(64)

Defining ,

(65)

which is a normal distribution. The binomial distribution is therefore approximated by a normal distribution for any fixed (even if is small) as is taken to infinity.

If infty" border="0" height="14" width="40"> and 0" border="0" height="14" width="33"> in such a way that lambda" border="0" height="14" width="46">, then the binomial distribution converges to the Poisson distribution with mean .

Let and be independent binomial random variables characterized by parameters and . The conditional probability of given that is

(66)

Note that this is a hypergeometric distribution.

Normal Distribution

2010-01-28T05:57:00.000-08:00

A normal distribution in a variate with mean and variance is a statistic distribution with probability density function

(1)

on the domain . While statisticians and mathematicians uniformly use the term "normal distribution" for this distribution, physicists sometimes call it a Gaussian distribution and, because of its curved flaring shape, social scientists refer to it as the "bell curve." Feller (1968) uses the symbol for in the above equation, but then switches to in Feller (1971).

de Moivre developed the normal distribution as an approximation to the binomial distribution, and it was subsequently used by Laplace in 1783 to study measurement errors and by Gauss in 1809 in the analysis of astronomical data (Havil 2003, p. 157).

The normal distribution is implemented in Mathematica as NormalDistribution[mu, sigma].

The so-called "standard normal distribution" is given by taking and in a general normal distribution. An arbitrary normal distribution can be converted to a standard normal distribution by changing variables to , so , yielding

(2)

The Fisher-Behrens problem is the determination of a test for the equality of means for two normal distributions with different variances.

The normal distribution function gives the probability that a standard normal variate assumes a value in the interval ,

			(3)
			(4)

where erf is a function sometimes called the error function. Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so both must be either computed numerically or otherwise approximated.

The normal distribution is the limiting case of a discrete binomial distribution as the sample size becomes large, in which case is normal with mean and variance

			(5)
			(6)

with .

The distribution is properly normalized since

(7)

The cumulative distribution function, which gives the probability that a variate will assume a value , is then the integral of the normal distribution,

			(8)
			(9)
			(10)

where erf is the so-called error function.

Normal distributions have many convenient properties, so random variates with unknown distributions are often assumed to be normal, especially in physics and astronomy. Although this can be a dangerous assumption, it is often a good approximation due to a surprising result known as the central limit theorem. This theorem states that the mean of any set of variates with any distribution having a finite mean and variance tends to the normal distribution. Many common attributes such as test scores, height, etc., follow roughly normal distributions, with few members at the high and low ends and many in the middle.

Because they occur so frequently, there is an unfortunate tendency to invoke normal distributions in situations where they may not be applicable. As Lippmann stated, "Everybody believes in the exponential law of errors: the experimenters, because they think it can be proved by mathematics; and the mathematicians, because they believe it has been established by observation" (Whittaker and Robinson 1967, p. 179).

Among the amazing properties of the normal distribution are that the normal sum distribution and normal difference distribution obtained by respectively adding and subtracting variates and from two independent normal distributions with arbitrary means and variances are also normal! The normal ratio distribution obtained from has a Cauchy distribution.

Using the k-statistic formalism, the unbiased estimator for the variance of a normal distribution is given by

(11)

where

(12)

(13)

The characteristic function for the normal distribution is

(14)

and the moment-generating function is

		" border="0" height="20" width="27">	(15)
			(16)
			(17)

			(18)
			(19)

and

			(20)
			(21)

These can also be computed using

			(22)
			(23)
			(24)

yielding, as before,

			(25)
			(26)

The raw moments can also be computed directly by computing the raw moments " border="0" height="17" width="52">,

(27)

(Papoulis 1984, pp. 147-148). Now let

			(28)
			(29)
			(30)

giving the raw moments in terms of Gaussian integrals,

(31)

Evaluating these integrals gives

			(32)
			(33)
			(34)
			(35)
			(36)

Now find the central moments,

			(37)
			(38)
			(39)
			(40)

The variance, skewness, and kurtosis excess are given by

			(41)
			(42)
			(43)

The cumulant-generating function for a normal distribution is

			(44)
			(45)

			(46)
			(47)
		2." border="0" height="14" width="62">	(48)

For normal variates, for 2" border="0" height="14" width="27">, so the variance of k-statistic is

			(49)
			(50)

Also,

			(51)
			(52)
			(53)

where

			(54)
			(55)

The variance of the sample variance for a general distribution is given by

(56)

which simplifies in the case of a normal distribution to

(57)

(Kenney and Keeping 1951, p. 164).

If is a normal distribution, then

(58)

so variates with a normal distribution can be generated from variates having a uniform distribution in (0,1) via

(59)

However, a simpler way to obtain numbers with a normal distribution is to use the Box-Muller transformation.

The differential equation having a normal distribution as its solution is

(60)

since

(61)

(62)

(63)

This equation has been generalized to yield more complicated distributions which are named using the so-called Pearson system.

The normal distribution is also a special case of the chi-squared distribution, since making the substitution

(64)

gives

			(65)
			(66)

Now, the real line is mapped onto the half-infinite interval by this transformation, so an extra factor of 2 must be added to , transforming into

			(67)
			(68)

(Kenney and Keeping 1951, p. 98), where use has been made of the identity . As promised, (68) is a chi-squared distribution in with (and also a gamma distribution with and ).

One and Two Sample Test of Hypothesis

2010-01-28T05:14:00.000-08:00

Statistical Decision: - Very often is practice; we are called upon to make decisions about population on the basic of sample information; such decisions are called Statistical Decision.

Statistical Hypothesis: - Such assumptions about the population may or may not be true are called Statistical Hypothesis.

Null Hypothesis: - If we want to decide whether one assumption or procedure is letter than another. Null Hypothesis is the positive rejection of Hypothesis. It is defined as Ho.

Alternative Hypothesis: - The Alternative Hypothesis is the logical opposite to the Null Hypothesis. It is defined as Ha.

One Tail Test: - A test of any Statistical Hypothesis where the alternative is one sided such as

Ho : μ = μ_o or Ho : μ = μ_o

Ha : μ > μ_o or Ha : μ < μ_o

Is called one Sided Test.

Two Tailed Test: A test of any Statistical Hypothesis where the alternative is two sided such as

Ho : μ = μ_o

Ha : μ ≠ μ_o

Is called Two Sided Test.

A test of the mean of a normal distribution against two sided alternative:

A test of the mean of a normal distribution against right sided alternative: