Combinations Functions of Random Variables Linear Functions of
Combinations & Functions of Random Variables
Linear Functions of a Random Variable n If X is a random variable and Y is a linear function of the random variable X, where Y = a. X +b (a & b are numbers) then it is a general rule that: E(Y) = a. E(X) +b and Var(Y) = a 2 Var(X) (how? ) 2
Linear Functions of a Random Variable n 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 3
Linear Combinations of Random Variables If X 1, X 2, ……. . , Xn is a sequence of random variables and a 1, a 2, …. . , an and b are constants, and Y is a linear combination in the following form Y = a 1 X 1 + a 2 X 2 + ……. . + an. Xn + b then n E(Y) = a 1 E(X 1) + a 2 E(X 2) + ……. . + an. E(Xn)+ b and Var(Y) = (a 1)2 Var(X 1)+ (a 2)2 Var(X 2)+…+ (an)2 Var(Xn) 4
Example: Suppose that X 1 and X 2 are two independent random variables and both of them have an expectation of μ and a variance of σ2. and suppose that Y 1 and Y 2 are two random variable for which Y 1 = X 1 + X 2 Y 2 = X 1 – X 2 Find the expectations and variances of Y 1 and Y 2. n E(Y 1)= E(X 1)+ E(X 2) = 2μ and E(Y 2)= E(X 1) – E(X 2) = zero Var(Y 1)= (1)2 Var(X 1)+ (1)2 Var(X 2) = 2σ2 and Var(Y 2)= (1)2 Var(X 1)+ (-1)2 Var(X 2) = 2σ2 5
Conclusion from the previous example: n Adding or subtracting independent random variables increases variability 6
Averaging Independent Random Variables n Suppose that X 1 , X 2, ……, Xn is a sequence of independent random variables each with an expectation μ and a variance of σ2 , and with an average of 7
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