Advanced High-School Mathematics

318 CHAPTER 6 Inferential Statistics

6.1 Discrete Random Variables

Let’s start with an example which is probably familiar to everyone. We
take a pair of fair dice and throw them, letting X be the sum of the
dots showing. Of course,X is random as it depends on the outcome
of the experiment. FurthermoreX is discrete: it can only take on the
integer values between 2 and 12. Finally, using elementary means it is
possible to compute the probability thatX assumes any one of these
values. If we denote byP(X=x) the probability thatXassumes the
valuex, 2 ≤x≤12 can be computed and tabulated as below:

x 2 3 4 5 6 7 8 9 10 11 12

P(X=x) 361 362 363 364 365 366 365 364 363 362 361

The table above summarizes the distributionof the discrete ran-
dom variable X. That is, it summarizes the individual probabilities
P(X=x), wherextakes on any one of the allowable values. Further-
more, using the above distribution, we can compute probabilities of the
formP(x 1 ≤X≤x 2 ); for example

P(2≤X≤5) =P(X= 2)+P(X= 3)+P(X= 4)+P(X= 5) = 361 + 362 + 363 + 364 =^1036.

It is reasonably clear that ifXis an arbitrary discrete random vari-
able whose possible outcomes arex 1 , x 2 , x 3 ,...,

∑∞

i=1

P(X=xi) = 1.

This of fundamental importance!

6.1.1 Mean, variance, and their properties

We define themeanμX (orexpectationE(X))^1 of the discrete ran-
dom variableXby setting

(^1) Some authors use the notation〈X〉for the mean of the discrete random variableX.