Discrete vs. Continuous Random Variables

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.

Statistics Tutorial: Attributes of Random Variables

Just like variables from a data set, random variables are described by measures of central tendency (i.e., mean and median) and measures of variability (i.e., standard deviation and variance). This lesson shows how to compute these measures for discrete random variables.

Mean of a Discrete Random Variable

The mean of the discrete random variable X is also called the expected value of X. Notationally, the expected value of X is denoted by E(X). Use the following formula to compute the mean of a discrete random variable.

E(X) = μ_x = Σ [ x_i * P(x_i) ]

where x_i is the value of the random variable for outcome i, μ_x is the mean of random variable X, and P(x_i) is the probability that the random variable will be outcome i.

Median of a Discrete Random Variable

The median of a discrete random variable is the "middle" value. It is the value of X for which P(X < x) is greater than or equal to 0.5 and P(X > x) is greater than or equal to 0.5.

Consider the problem presented above in Example 1. In Example 1, the median is 2; because P(X < 2) is equal to 0.60, and P(X > 2) is equal to 0.70. The computations are shown below.

P(X < 2) = P(x=0) + P(x=1) + P(x=2) = 0.10 + 0.20 + 0.30 = 0.60

P(X > 2) = P(x=2) + P(x=3) + P(x=4) = 0.30 + 0.25 + 0.15 = 0.70

Variability of a Discrete Random Variable

The standard deviation of a discrete random variable (σ) is equal to the square root of the variance of a discrete random variable (σ²). The equation for computing the variance of a discrete random variable is shown below.

σ² = Σ [ x_i - E(x) ]² * P(x_i)

where x_i is the value of the random variable for outcome i, P(x_i) is the probability that the random variable will be outcome i, E(x) is the expected value of the discrete random variable x.