Need help with an expansion/simplification problem...please....?

Question

σ^2=Σ(x-μ)^2*P(x)
becomes
Σx^2*P(x)-μ

My question is...how??? I'm not sure how to expand and simplify to get the answer. If anyone could show me I would greatly appreciate it!

Answer 1

This is a really neat expansion that once you've seen it, you'll always remember:

Σ(x-μ)^2 P(x)
= Σ(x^2 - 2μx + μ^2) P(x)
= Σx^2 P(x) - 2μ ΣxP(x) + μ^2 ΣP(x)
= Σx^2 P(x) - 2μ (μ) + μ^2 (1)
since by definition μ = ΣxP(x) and ΣP(x) = 1
= Σx^2 P(x) - μ^2.