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The Normal Distribution
Definition. If X has density
f(x) =
{ }
1 −(x − µ)
2
√ exp 2πσ
2 2σ 2
then X has the normal distribution, and we
write X ∼ N(µ, σ 2 ).
• E(X) = µ and Var(X) = σ 2 .
• If Z ∼ N(0, 1) then Z is standard normal.
Proposition. Let Z ∼ N(0, 1) and set
X = µ + σZ for constants µ and σ. Then,
X ∼ N(µ, σ 2 ).
Proof.
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