Question

Suppose X, Y and Z are independent standard normal random variables. Then W = 2X + Y - Z is a random variable with mean 0 and variance 2, but not necessarily normal distributed. a normal random variable with mean 0 and variance 4. O a random variable with mean 0 and variance 4, but not necessarily normal distributed. a random variable with mean 0 and variance 6, but not necessarily normal distributed. a normal random variable with mean 0 and variance 2. a normal random variable with mean 0 and variance 6.

Answer 1

X, Y and Z are independent standard normal variables.

Here X, Y and Z are independent so all covariance terms are 0.

X ~ N(0,1) ; Y ~ N(0,1) and Z~ N(0,1)

W = 2X + Y - Z

E(W) = E(2X + Y - Z) = 2* E(X) + E(Y) - E(Z) = 2*0+0-0 = 0

Mean of W is 0

V(W) = V(2X + Y + Z)

= ...{all covariance 0}

22 * V(X) +V(Y) +V(Z)

= 4*1 + 1 + 1

= 6

Variance of W is 6

Ans.:- a normal random variable with mean 0 and variance 6.