Introduction to Probability and Statistics for Engineers and Scientists

6.5Sampling Distributions from a Normal Population 215

6.5.1 Distribution of the Sample Mean

Since the sum of independent normal random variables is normally distributed, it follows
thatXis normal with mean

E[X]=

∑n

i= 1

E[Xi]

n

=μ

and variance

Var(X)=

1

n^2

∑n

i= 1

Var(Xi)=σ^2 /n

That is,X, the average of the sample, is normal with a mean equal to the population mean
but with a variance reduced by a factor of 1/n. It follows from this that

X−μ

σ/

√

n

is a standard normal random variable.

6.5.2 Joint Distribution ofXandS^2

In this section, we not only obtain the distribution of the sample varianceS^2 , but we also
discover a fundamental fact about normal samples — namely, thatXandS^2 are indepen-
dent with (n−1)S^2 /σ^2 having a chi-square distribution withn−1 degrees of freedom.
To start, for numbersx 1 ,...,xn, letyi=xi−μ,i=1,...,n. Then asy=x−μ,
it follows from the identity

∑n

i= 1

(yi−y)^2 =

∑n

i= 1

y^2 i−ny^2

that

∑n

i= 1

(xi−x)^2 =

∑n

i= 1

(xi−μ)^2 −n(x−μ)^2

Now, ifX 1 ,...,Xnis a sample from a normal population having meanμvarianceσ^2 ,
then we obtain from the preceding identity that

∑n

i= 1

(Xi−μ)^2

σ^2

=

∑n

i= 1

(Xi−X)^2

σ^2

+

n(X−μ)^2

σ^2