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.

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