Expectation and Variance

4.2.2 Expectation and Variance

Let c be a constant, for example let c = 5.

∑ Var (X ) = pi[(xi - μx)2] = σ2x

$∑ ∑ ∑ V ar(cX ) = pi[(c(xi)- cμx)2] = pi[c((xi)- μx )2] = c2 pi[((xi)- μx)2] = c2σ2x$

∑ 2 ∑ 2
V ar(5X ) = pi[(5(xi) - 5μx ) ] = pi[5((xi) - μx) ]
2∑ 2 2 2 2
= 5 pi[((xi) - μx) ] = 5 σ x = 25σx

∑ ∑
V ar(X + c) = pi[((xi + c) - (μx + c))2] = pi[((xi) + c - μx - c)2]
∑
= pi[((xi) - μx)2] = σ2
x

∑ ∑
V ar(X + 5) = pi[((xi + 5) - (μx + 5 ))2] = pi[((xi) + 5 - μx - 5)2]
∑
= pi[((xi) - μx)2] = σ2x