5.6 Normal distribution

Theorem 5.6.7 If the random variables X1,...,Xk are independent and if Xi has the normal distribution with mean μi and variance σi ^2, (i = 1, . . . , k), then the sum X1 + ... + Xk has the normal distribution with mean μ1 + ... + μk and variance σ1^2  + ... + σk2

Corollary 5.6.1 If the random variables X1,...,Xk are independent, if Xi has the normal distribution with mean μi and variance σ2 i (i = 1, . . . , k), and if a1,...,ak and b are constants for which at least one of the values a1,...,ak is different from 0, then the variable a1X1 + ... + akXk + b has the normal distribution with mean a1μ1 + ... + akμk + b and variance a2 1σ2 1 + ... + a2

Corollary 5.6.2 Suppose that the random variables X1,...,Xn form a random sample from the normal distribution with mean μ and variance σ^2, and let Xn denote their sample mean. Then Xn has the normal distribution with mean μ and variance σ^2/n.