Problem 5. (2 points) If X,, X, ..X40 are independent random variables with means u, =...

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Question

Problem 5. (2 points) If X,, X, ..X40 are independent random variables with means u, = 2 = ... = H40 = 1, and variances o? =

math Statistics-And-Probability

Answer 1

Answer #1

The mean and variance of Y here is computed as:

My = E(2X1 - 3X2 + X3 + X4+ ... + X40)

My = 2px; – 3ux, + px3 + 4x4 + ... + HX40

uy = 2*1-3*1+1+1+1.....(1 is 38times)

ly = -1+1 + 38 = 37

therefore 37 is the mean of Y here.

The variance of Y here is computed as:
Note that as all X_i are independent, therefore we have here:

Var(Y) = Var (2X1 – 3X2 + X3 + X4+ ..... + X 40)

Var(Y) = 2°Var(x1) + (-3) Var (X2) + Var(X3) + Var(X4) + ... + Var (X40)

Var(Y) = 4Var(X1) +9Var (X2) +Var (X3) +Var(x4) + ... +Var X 40)

Var(Y) = 4*1+9+1+1*38 = 51

Therefore 51 is the required variance here.

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Answer 2

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