STAT 400 Homework 8 | Complete solution

STAT 400 G.  Homework 8

Readings
Sections 3.3 (Normal distribution), See also Tables Va and Vb in pg. 494-495.
Section 4.1 (Joint distribution and Independence)
Section 4.3 (Conditional distributions)
Exercises
1. Linear Transformations of Normal Random Variables are also Normal
(a) Let X N(; 2) and a; b 2 R. Show that Y = aX + b is also a normal random
variable. With what parameters?
Hint: Compute FY (y) = P(Y y) = P(X ?) =
R
: : :.
(b) Suppose that the temperature in some town on some randomly chosen day follows
the normal distribution with mean 15 and standard deviation 5, when measured
in oC. What can you say about the temperature in oF?
Hint: oF = 32 + (9=5) oC:
2. Moment-generating function of the normal
(a) Let Z N(0; 1) and show that
MZ(t) = E[expftZg] = exp

t2
2

; t 2 R:
Hint: This formula is obtained with a technique called "completion of a square",
which is often used with Gaussian random variables. See pg. 106.
(b) Let X N(; 2) and show that
MX(t) = E[expftXg] = exp

t +
2
2
t2

; t 2 R:
Hint: Use (a) and the fact that X can be expressed as X = + Z, where
Z N(0; 1).
1
STAT 400 G. Fellouris Fall 2015
(c) Ex. 3.6 from Section 3.3.
3. Percentiles of the normal distribution
Recall that the (100p)th percentile of a (continuous) random variable with cdf F is the
real number xp such that F(xp) = p.
(a) Show that
(100p)th percentile of N(; 2) = + (100p)th percentile of N(0; 1).
(b) Let Z N(0; 1) and 2 (0; 1). We denote by z the (100(1 􀀀 ))th percentile
of Z, that is the real number that satis es P(Z > z ) = : This notation will be
extensively used in Statistics. See pg. 109.
Show that 􀀀z is the (100 )th percentile of N(0; 1) and nd z for = 10%; 5%; 1%.
(c) Suppose that the LSAT score of a randomly chosen entering student in a law
school class is normally distributed with mean 160 and standard deviation 10.
What is the LSAT score that a randomly chosen student should have in order to
be in the top 10%?
4. Ex. 11 from Section 3.3.
5. Ex. 13 from Section 3.3.
6. Ex. 15 from Section 3.3.
7. A nancial contract
Let St be the dollar price of a stock at t years from today. Consider an agreement
(contract) between two parties, the \holder" and the \seller", according to which the
seller pays the holder $1 at time t if St > K, where K is some positive constant. If
St K, no transaction occurs,
(a) Show that the fair value of this contract, that is the amount the seller should pay
the holder to convince him/her to enter this contract, is P(St > K).
2
STAT 400 G. Fellouris Fall 2015
(b) Suppose that St is a lognormal random variable, i.e.,
ln

St
S0

N

􀀀
2
2
t; 2t

;
where S0 is the price of the stock today, and are positive numbers. Obtain
an expression for P(St > K) that involves only S0;K; ; ; t and the cdf of the
standard normal distribution.
(c) Compute this probability when S0 = 60;K = 70; = 0:3; = 0:2; t = 0:5.
8. The sum of two independent Poissons is Poisson
Let X; Y be two independent Poisson random variables with parameters and ,
respectively.
(a) Show that X +Y is also Poisson with parameter +, in other words, show that
P(X + Y = k) = e􀀀(+) ( + )k
k!
; k = 0; 1 : : :
Hint: Start with the observation that
fX + Y = kg =
[k
m=0
fX = m; Y = k 􀀀 mg
and apply the addition rule. Recall also the Binomial expansion.
(b) Show that the conditional pmf of X given X+Y = n is Binomial with parameters
n and p = =( + ), i.e.,
P(X = kjX + Y = n) =

n
k


+
k

+
n􀀀k
; k = 0; : : : ; n:
(c) What is E[XjX + Y = n] and Var[XjX + Y = n]?
9. Ex. 4 in Section 4.1
10. Ex. 6 in Section 4.3