322 CHAPTER 6 Inferential Statistics
As you might expect, the above formula isfalsein general (i.e., when
XandY not independent); see Exercise 1, below. Using (6.5), we see
immediately that ifXis a discrete random variable, and ifY =aX+b,
wherea andbare real numbers, then we may regardbas a (constant)
random variable, certainly independent of the random variable aX.
Therefore,
Var(Y) = Var(aX+b) = Var(aX) + Var(b) =a^2 Var(X),where we have used the easily-proved facts that Var(aX) =a^2 Var(X)
and where the variance of a constant random variable is zero (see Ex-
ercises 5 and 6, below).
We conclude this section with a brief summary of properties of mean
and variance for discrete random variables.^5
- IfX is a random variable, and ifa, bare real numbers, then
E(aX+b) =aE(X) +b. - IfX is a random variable, and ifa, bare real numbers, then
Var(aX+b) =a^2 Var(X). - IfX andY are random variables, then
E(X+Y) =E(X) +E(Y). - IfX andY areindependentrandom variables, then
E(XY) =E(X)E(Y). - IfX andY are independent random variables, then
Var(X+Y) = Var(X) + Var(Y).
6.1.2 Weak law of large numbers (optional discussion)
In order to get a better feel for the meaning of the variance, we include
the following two lemmas:
(^5) These same properties are also true for continuous random variables!