Question

If a random variable X1 has mean 4 and variance 9, while the random variable X2
has mean − 2and variance 5, and the two are independent, find
i) E(2X1+ X2- 5)
ii) Var(2X1+ X2- 5)

Answer 1

i) For random variable X:

$$E(aX+b)=aE(x)+b$$

For random variables X1, X2

$$E(X1+X2)=E(X1)+E(X2)$$

Then

$$E(2X1+X2-5)=2E(X1)+E(X2)-5=2(4)-2-5=1$$

$$E(2X1+X2-5)=2(4)-2-5=1$$

ii) For random variable X:

$$Var(aX+b)=a^2Var(X)$$

If X1, X2 are independent, then

$$Var(X1+X2)=Var(X1)+Var(X2)$$

Then

$$Var(2X1+X2-5)=2^2 Var(X1)+Var(X2)$$

$$Var(2X1+X2-5)=4(9)+5=41$$