Let X = aZ + b, where Z is a standard normal random variable, and a, b are two unknown constants. It is given that

E[...

Question

Let X = aZ + b, where Z is a standard normal random variable, and a, b are two unknown constants. It is given that

E[X] = 1,  E[(X - E[X])^2] = -2,  E[(X - E[X])^2]^2 = 4

where E[X] denotes the expectation of random variable X. The values of a, b are:

A. a = -2, b = 1
B. a = 2, b = -1
C. a = -1, b = 3
D. a = 1, b = 0.25

Accepted Answer

A. a = -2, b = 1 If Z is a standard normal random variable (mean 0, variance 1), then X = aZ + b is a normal random variable with mean b and variance a^2. The condition E[X] = 1 gives b = 1. The variance condition Var(X) = a^2 = 4 gives a = +2 or a = -2. The additional condition (or the answer choices) uniquely selects a = -2, b = 1 as option A. Note that the second condition as written E[(X-E[X])^2] = -2 cannot literally equal -2 since variance is non-negative; the GATE question likely states Var(X) = 4 or uses a different expression, and the answer is determined from the constraint a^2 = 4 with b = 1.