Mean and Standard Deviation

of Discrete Random Variables

Discrete random variables can be written as probability distributions:

X x1 x2 x3 … xn

P(X) p1 p2 p3 … pn

xi represents the ith outcome and pi is its probability of occurrence.

X x1 x2 x3 … xn

P(X) p1 p2 p3 … pn

For any discrete random variable X, with n possible outcomes we can define the mean, X, or expected value of X, E(X), to be the weighted average of the probability distribution. This means that outcomes that occur more often have more influence on the mean.

We compute this mean by multiplying the outcome by the probability for each outcome, then summing all products.

X x1 x2 x3 … xn

P(X) p1 p2 p3 … pn

μX =x1p1+x2p2+x3p3+...+xnpn

μX = xii=1

n

∑ pi

or

X x1 x2 x3 … xn

P(X) p1 p2 p3 … pn

We define the variance of X as follows: Subtract the mean from each outcome, square the difference, multiply by the probability, and sum all products.

σX2 =(x1−μX)

2p1+(x2−μX)2p2+(x3−μX )

2p3+...+(xn −μX)2pn

σX2 = (xi

i=1

n

∑ −μX)2pi

or

The standard deviation is the square root of the variance.

σX = (xii=1

n

∑ −μX )2pi

Each of these tasks can be easily accomplished using the graphing calculator.

X 0 1 23 4

P(X) .1 .3 .2 .3 .1

Consider this probability distribution for the random variable X:

Enter the X values in L1 and the probabilities in L2.

Perform 1-Var Stats L1,L2.

The calculator results are shown:

The mean is 2, and the standard deviation is 1.1832.

Note that the standard deviation in this case is a population standard deviation and so is written , not s. The calculator has calculated only. Recall that s has an n-1 quantity in the denominator. Probability distributions will always have n=1 and so will not have an s calculated unless there is a data error. (When n=1, the denominator goes to 0.)

The end