Question

5. A random variable X ∼ N (µ, σ2 ) is Gaussian distributed with mean µ and variance σ 2 . Given that for any a, b ∈ R, we have that Y = aX + b is also Gaussian, find a, b such that Y ∼ N (0, 1)

Please show your work. Thanks!

Answer 1

E(X) = μ, Var(X) = σ^2

Y = aX + b

We want E(Y) = 0 and Var(Y) = 1

E(Y) = a E(X) + b

0 = a μ + b

b = -a μ --- (1)

Var(Y) = (a^2) Var(X)

1 = (a^2) σ^2

a^2 = 1/σ^2

a = ± 1/σ --- (2)

From (1) and (2)

b = (± 1/σ) μ

So, a = ± (1/σ) and b = ± (μ/σ)