Question
Let X1, X2 be two independent normal random variables with means µ1, µ2 and standard deviations σ1, σ2, respectively. Consider Y = X1 – X2; µ1=µ2 =1, σ1=1,σ2= 2. Then,
Options :
Y is normally distributed with mean 0 and variance 1
Y is normally distributed with mean 0 and variance 5
Y has mean 0 and variance 5, but is NOT normally distributed
Y has mean 0 and variance 1, but is NOT normally distributed
Answer :
Y is normally distributed with mean 0 and variance 5
Solution :
µ1=µ2 =1, σ1=1,σ2= 2
x1 and x2, are two independent random variables
Y=X1-X2
µ (Y)=µ(X1-X2 )
= µ(X1)-µ(X2 ) = µ1-µ2 = 1-1 = 0
Var (Y) = Var (X1 — X2)
= Var (X1) + Var (X2) - Cov(X1,X2)
Since X1, and X2, are independent variables
Var(Y) = Var(X1) + Var(X2)
= σ12+ σ22 = 1+4
Var(Y) =
Copyright © 2025 Test Academy All Rights Reserved