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
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 jo
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 . Sho
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 .
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
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 ,
w
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 ii
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 m
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.
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) =
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 | ) =
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 distrib
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
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),
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, cou
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;
thu
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
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) L
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 ar