Description

1 Definitions
1. The pmf for a Bernoulli r.v. Xi is given by

p
10− p0 kk == 10
pXi(k) = 0 otherwise
2. The pmf for a Binomial Random variable (1.1)
n
X
X = Xi
i=1
is given by (1.2)
!
n k n−k pX (X = k) = p0 (1 − p0) , 0 ≤ k ≤ n
k
where Xi are i.i.d.
3. Let Z ∼ N(0,1). Then the Q function is defined as, (1.3)
Q(z) = Pr(Z > z), z ≥ 0
4. The MGF of Y is defined as (1.4)

MY(s) = E e−sY
5. The MGF of Z is (1.5)
1 1
MZ(s) = e2 s2
2
6. (1.6)
!n
1
lim 1 + = e
n→∞ n
2 Problems (1.7)
1. Show that the mean and variance of Xi are µ = p0 and σ2 = p(1 − p).
2. Show that the mean and variance of
Xi − µ Yi = σ (2.1)
are 0 and 1 respectively. 3. Show that the mean and variance of
n
1 X
Y = √ Yi (2.2)
n i=1
are 0 and 1 respectively.
4. Show that

− sYi n
√

MY(s) = E e n (2.3)
5. Show that
sY s2 1
E e= 1 + + R(s,n) (2.4)
2n n
where R(s,n) is an infinite series. 6. Show that
lim Y = Z (2.5)
n→∞
This is known as the Central Limit Theorem. 7. Let
n
1 X
p = Xi (2.6) n
i=1
Show that
E (p) = µ = p0 (2.7) σ2 p0 (1 − p0)
var (p) = = (2.8) n n
8. Show that
p − p0
lim= Z (2.9)
n→∞ q p0(1−p0) n
9. Find z if
Pr(|Z| < z) = 1 − α (2.10)
10. Show that
rr
p0 − Q < p < p0 + 0 (1 − p0)Q−1 α (2.11) p0 (1 − p0) −1 α p
n 2 n 2
11. Among 4000 newborns, 2080 are male. Find the 1 − α = 0.99 confidence interval of the probability that a male child is born.