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!
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)
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