1. Combinations & Functions of Random Variables

2. Linear Functions of a Random Variable
If X is a random variable and Y is a linear function of the random variable X, where
Y = aX +b (a & b are numbers)
then it is a general rule that:
E(Y) = aE(X) +b
and Var(Y) = a2 Var(X) (how?)

3. Linear Functions of a Random Variable
An important application of this result will be used in chapter 5, which concerns the “standardization” of a random variable X to have a zero mean and a unit variance. The new “standardized” random variable will be:
If you apply the previous linear function rule, then

4. Linear Combinations of Random Variables
If X1, X2, …….. , Xn is a sequence of random variables and a1, a2, ….., an and b are constants, and Y is a linear combination in the following form
Y = a1X1 + a2X2 + …….. + anXn + b
then
E(Y) = a1E(X1) + a2E(X2) + …….. + anE(Xn)+ b
and
Var(Y) = (a1)2Var(X1)+ (a2)2Var(X2)+…+ (an)2Var(Xn)

5. Example:
Suppose that X1 and X2 are two independent random variables and both of them have an expectation of μ and a variance of σ2 .
and suppose that Y1 and Y2 are two random variable for which
Y1 = X1 + X Y2 = X1 – X2
Find the expectations and variances of Y1 and Y2.
E(Y1)= E(X1)+ E(X2) = 2μ and E(Y2)= E(X1) – E(X2) = zero
Var(Y1)= (1)2Var(X1)+ (1)2Var(X2) = 2σ2 and
Var(Y2)= (1)2Var(X1)+ (-1)2Var(X2) = 2σ2

6. Conclusion from the previous example:
Adding or subtracting independent random variables increases variability

7. Averaging Independent Random Variables
Suppose that X1 , X2, ……, Xn is a sequence of independent random variables each with an expectation μ and a variance of σ2 , and with an average of