3.2 - Mean, also called Expected Value, of a Discrete Variable

The phrase expected value is a synonym for mean value in the long run (meaning for many repeats or a large sample size). For a discrete random variable, the calculation is Sum of (value × probability) where we sum over all values (after separately calculating value × probability for each value), expressed as:

\( E(X)=\sum x_i p_i\),

meaning we take each observed X value and multiply it by its respective probability. We then add these products to reach our expected value labeled E(X). [NOTE: the letter X is a common symbol used to represent a random variable. Any letter can be used.]

Example : A fair six-sided die is tossed. You win \$2 if the result is a “1”, you win \$1 if the result is a “6” but otherwise you lose \$1.

\( Expected\ Value= (2 \times \frac {1}{6})+(1 \times \frac {1}{6})+(-1 \times \frac {4}{6})= - \frac {1}{6}=-\$ 0.17 \)

The interpretation is that if you play many times, the average outcome is losing 17 cents per play.

Example : Using the probability distribution for number of tattoos given above (not the cumulative!),

The mean number of tattoos per student is

\( Expected\ Value=(0 \times 0.85)+(1 \times 0.12)+ (2 \times 0.015) +(3 \times 0.010) +(4 \times 0.005) =0.20 \)

Standard Deviation of a Discrete Variable

Knowing the expected value is not the only important characteristic one may want to know about a set of discrete numbers: one may also need to know the spread, or variability, of these data. For instance, you may "expect" to win \$20 when playing a particular game (which appears good!), but the spread for this might be from losing \$20 to winning \$60. Knowing such information can influence you decision on whether to play.

To calculate the standard deviation we first must calculate the variance. From the variance, we take the square root and this provides us the standard deviation. Your book provides the following formula for calculating the variance:

\( \sigma ^2= \sum (x_i-\mu)^2 p_i \) and the standard deviation is:\( \sigma = \sqrt {\sum (x_i-\mu)^2 p_i}\)

In this expression we substitute our result for E(X) into u , and u is simply the symbol used to represent the mean of some population .

However, an easier formula to use and remember for calculating the standard deviation is the following:

\( \sigma ^2= \sum x_i^2 p_i -\mu ^2\) and again we substitute E(X) for μ.

The standard deviation is then found by taking the square root of the variance. Notice in the summation part of this equation that we only square each observed X value and not the respective probability.

Example : Going back to the first example used above for expectation involving the die, we would calculate the standard deviation for this discrete distribution by first calculating the variance:

\( \sigma ^2= \sum x_i^2 p_i -\mu ^2 = (2^2 \times \frac{1}{6})+(1^2 \times \frac{1}{6})+(-1)^2 \times \frac{4}{6}-(- \frac{1}{6})^2\)

\(= \frac{4}{6}+\frac {1}{6}+ \frac{4}{6}-\frac{1}{36} = \frac{53}{36}=1.472 \)

So the standard deviation would be the square root of 1.472, or 1.213