Stat 571 Discussion #4 Fall, 2003
Review
1. The mean and variance of X∼Binomial(n,p) are E(X) = np, V ar(X) = np(1 −p).
2. If X1,X2, ..., Xnconstitute a random sample from a distribution with mean µand variance
σ2, then:
E(¯
X) = µand V ar(¯
X) = σ2/n
E(
n
X
i=1
Xi) = nµ and V ar(
n
X
i=1
Xi) = nσ2
3. Let random sample X1, X2,···, Xn∼N(µ, σ2), then random variables ¯
Xand Pn
i=1 Xiare
distributed as
¯
X∼N(µ, σ2
n) and
n
X
i=1
Xi∼N(nµ, nσ2).
4. Central Limit Theorem
For a random sample X1, X2,···, Xn(when n is large) from an arbitrary distribution with
mean µand variance σ2,¯
Xwill be approximately distributed by ¯
X∼N(µ, σ2
n) .
5. Let X∼B(n, p), then the normal approximation for Xis
X∼N(np, np(1 −p)).
The proportion Y=X
ncan be approximated by a normal distribution N(p, p(1 −p)/n).
The requirement is that np ≥5 and n(1 −p)≥5 for both approximations.
6. Let random sample X1, X2,···, Xn∼N(µ, σ2), then random variable
V2=(n−1)S2
σ2∼χ2
n−1,
where S2is the sample variance defined as S2=1
n−1Pn
i=1(Xi−¯
X)2.
Practice Problem
1. Suppose the random variable X is distributed as N(100,144).
(a) Consider a random sample with n= 16, what is the distribution of ¯
X=1
16 P16
i=1 Xi?
(b) Find out Pr( ¯
X < 103).
(c) What is the distribution of P16
i=1 Xi?
(d) How large does nhave to be in order that V ar(¯
X) = 1 ?
