Chapter 1: Probability
1
Introduction
The mathematical theory of probability has been applied to a wide variety
of phenomena. Here are some examples.
Probability theory has been used in genetics as a model for mutations and ensuing natural variability, a
STAT414, Fall Semester of 2013
Assignment 5
(Deadline: 10/08/2013)
1. (EX3.3) Three players play 10 independent rounds of a game, and each player has probability
1/3 of winning each round. Find the joint distribution of the numbers of games won by each
of
STAT414, Fall Semester of 2013
Assignment 6
(Deadline: 10/15/2013)
1. (EX3.64) Find the joint density of X + Y and X/Y , where X and Y are independent exponential random variables with parameter . Show that X + Y and X/Y are independent.
2. (Ex3.79) If T1
STAT414, Fall Semester of 2013
Assignment 7
(Deadline: 10/22/2013)
1. (4.30) Find E [1/(X + 1)], where X is a Poisson random variable.
2. (Ex4.32) Let X have a gamma distribution with parameters and . For those values of
and foe which it is dened, nd E (
STAT414, Fall Semester of 2013
Assignment 8
(Deadline: 10/29/2013)
1. (4.73) A fair coin is tossed n times, and the number pf heads, N , is counted. The coin is then
counted N times. Find the expected total number of heads generated by this process.
2. (E
STAT414, Fall Semester of 2013
Assignment 9
(Deadline: 11/05/2013)
1. (Ex4.89) Let X1 , . . . , Xn are independent normal random variables with means i and vari2
ances i . Show that Y =
n
i=1 i Xi ,
where the i are scalars, is normally distributed, and
nd
STAT414, Fall Semester of 2013
Assignment 10
(Deadline: 11/12/2013)
1. (Ex6.5) Show that if X Fn,m , then X 1 Fm,n .
2. (Ex6.6) Show that if T Tn , then T 2 F1,n .
3. (Ex6.9) Let X1 , . . . , Xn be iid random variables drawn from N (, 2 . Dene S 2 =
1
n1
STAT414, Fall Semester of 2013
Assignment 11
(Deadline: 11/19/2013)
1. (Ex8.7) Suppose that X follows a geometric distribution,
P (X = k ) = p(1 p)k1
and assume an iid sample of size n.
(a) Find the method of moments estimate of p.
(b) Find the MLE of p.
STAT414, Fall Semester of 2013
Assignment 4
(Deadline: 10/01/2013)
1. (EX2.33) Let F (x) = 1 exp(x ) for x 0, > 0, > 0, and F (x) = 0 for x < 0. Show
that F is a cdf, and nd the corresponding density.
2. (Ex2.38) If f and g are densities, show that f + (1
STAT414, Fall Semester of 2013
Assignment 3
(Deadline: 09/24/2013)
1. (EX2.1) Suppose that X is a discrete random variable with P (X = 0) = 0.25, P (X = 1) =
0.125, P (X = 2) = 0.125, and P (X = 3) = 0.5. Graph the frequency function and the
cumulative di
Chapter 5: Limit Theorems
1
The Law of Large Numbers
Denition 1.1 A sequence of random variables, X1 , X2 , . . ., converges in probability
to a random variable X if, for every > 0,
lim P (|Xn X | ) = 0
n
or equivalently,
lim P (|Xn X | < ) = 1.
n
The X1
Chapter 6: Distributions Derived from the Normal
Distribution
1
2 , t and F Distributions
Denition 1.1 If Z is a standard normal random variable, the distribution of U
is called the chi-square distribution with 1 degree of freedom.
If X
= Z2
N (, 2 ), th
Ch8 Estimation of Parameters and Fitting of
Probability Distributions
1
Introduction
Denition 1.1 A point estimator is any function W (X1 , . . . , Xn ) of a sample; that is,
any statistic is a point estimator.
Note that an estimator is a function of the
Chapter 4: Expected Values
1
The expected value of a random variable
Denition 1.1 If X is a discrete random variable with frequency function
expected value of X , denoted by E (X ), is
E (X ) =
p(x), the
xi p(xi )
i
provided that
i
|xi |p(xi ) < . If the
Chapter 3: Joint Distributions
1
Introduction
This chapter is concerned with the joint probability structure of two or
more random variables dened on the same sample space.
In ecological studies, counts of several species, modeled as random variables, ar
Chapter 2: Random Variables
1
Discrete Random Variables
As motivation for a denition, consider the following example. A coin is
thrown three times, and the sequence of heads and tails is observed;
thus,
= cfw_hhh, hht, htt, hth, ttt, tth, thh, tht.
Examp
STAT414, Fall Semester of 2013
Assignment 2
(Deadline: 09/17/2013)
1. (EX1.48) An urn contains three red and two white balls. A ball is drawn, and then it and
another ball of the same color are placed back in the urn. Finally, a second ball is drawn.
(a)
STAT414, Fall Semester of 2013
Assignment 1
(Deadline: 9/10/2013)
1. (Ex1.2) Two six-sided dice are thrown sequentiallu, and the face values that come up are
recorded.
(a) List the sample space.
(b) List the elements that make up the following events: (1)
STAT610, Fall Semester of 2013
Assignment 9
(Deadline: 11/05/2013)
1. (Ex4.22) Let (X, Y ) be a bivariate random vector with joint pdf f (x, y ). let U = aX + b and
V = cY + d, where a, b, c, and d are xed constants with a > 0 and c > 0. Show that the
joi