Measure, Integration & Real Analysis, 2021a

390 Chapter 12 Probability Measures
12.22 variance of sum of independent random variables
Suppose (Ω, F, P) is a probability space and X 1 ,...,X n ∈L 2 (P) are independent
random variables. Then
Proof
σ 2( ∑
n )
X k = E
k=1
( n
=E ∑
k=1
σ 2 (X 1 + ···+ X n )=σ 2 (X 1 )+···+ σ 2 (X n ).
Using the variance formula given by 12.20,wehave
( ( n ) ) ( 2
∑ X k − E ( n )
∑
) 2
X k
k=1 k=1
) (
2
X k + 2E
∑
1≤j