Suppose X is a normally distributed random variable with mean µ and variance Consider a new random variable, W=2X + 3.

Question

i. What is E(W)? ii. What is Var(W)?
 
Now suppose the random variables X and Y are jointly distributed.  Define a new random variable, W=2X+3Y.   iii. What is Var(W)? iv. What is Var(W) if X and Y are independent?

Answer 1

I only have solutions to parts (i) and (ii).

W=2X+3.

(i) E(W)=2µ+3, because all X data are shifted by the same amount in dataset W, so the mean is similarly shifted:

X={ x₁ x₂ x₃ ... }, µ=∑x/n

W={ 2x₁+3 2x₂+3 2x₃+3 ... }, E(W)=∑(2x+3)/n=2∑x/n+3n/n=2µ+3.

(ii) Var(W)=4Var(X)=4σ²:

Var(X)=∑(x-µ)²/n=(1/n)[∑x²-2µ∑x+nµ²]=

(1/n)[∑x²-(2µ)(nµ)+nµ²]=(1/n)[∑x²-nµ²]=

∑x²/n-µ². 

Var(W)=∑w²/n-(2µ+3)²=

∑(2x+3)²/n-(2µ+3)²=

(4∑x²+12∑x+9n)/n-(4µ²+12µ+9)=

(4∑x²+12nµ+9n)/n-(4µ²+12µ+9)=

4∑x²/n+12µ+9-4µ²-12µ-9=

4(∑x²/n-µ²)=4Var(X)=4σ².