Quiz 1
NAME: SOLUTIONS!
1. (6 pts) Let f (x) =
n
P
x
0
n! . Find f (2)
n=0
Solution:
n
P
x
x
n! is the taylor series expansion for e , in otherwords
n=0
X
xn
n=0
n!
= ex
Therefore,
f (x) = e2
f 0 (x) = ex
f 0 (2) = e2
2. (6 pts) Find
Solution:
Z x
d R
Homework 4
Solutions!
1. Let X Bin(n, p) where n is a positive integer larger than 1 (we have already
shown the case when n = 1) and 0 < p < 1. Derive the varaince of X (i.e. show
that V [X] = np(1 p)
Solution:
V [X] = E[X 2 ] E 2 [X]
also,
E[(X(X 1)] = E
EXAM 1
Please read all of the following information before starting the exam:
You are allowed to use One page for a cheat sheet, each double sided.
Answer each question completely. Show all work, clearly and in order, if you want to get full credit.
I
Outline
Section 3.7 - 3.9
Discrete Distributions Summary
STAT 3375Q
Chapter 3.7-3.9
Steven Chiou
University of Connecticut
09/20/2012
Steven Chiou
STAT 3375Q Chapter 3.7-3.9
University of Connecticut
Outline
Section 3.7 - 3.9
Discrete Distributions Summar
STAT 3375Q: Introduction to Mathematical Statistics
Kun Chen
Assistant Professor
Department of Statistics
University of Connecticut
Storrs, CT
August 22, 2013
Outline I
1
Introduction
2
Chapter 1
Introduction
What is Statistics
Statistics is the science o
3.107
The random variable Y follows a hypergeometric distribution with N = 6, n = 2, and r
= 4.
3.127
Let Y = # of typing errors per page. Then, Y is Poisson with = 4 and P(Y 4) = .6288.
Homework 2
Solutions
For multiple Solutions we will use the following Theorem:
Theorem 1 (Compliment Rule). Let A be an event in sample space S. Then P (AC ) =
1 P (A)
Proof.
P (S) = 1 Second Rule of Probability
P (S) = P (A AC ) Definition of Compliment
Homework 3
SOLUTIONS!
1. Problem 2.62 from the book (p 50)
Solution:
We can think of this simply in terms of the first production line. There are 22 71
ways to select three motors, two of which are from the particular supplier. Any
order of selecting th
STAT 3375Q: Introduction to Mathematical Statistics
Kun Chen
Assistant Professor
Department of Statistics
University of Connecticut
Storrs, CT
November 21, 2013
Outline I
1
Chapter 6 Function of Random Variables
6.3 The Method of Distribution Functions
6.
STAT 3375Q: Introduction to Mathematical Statistics
Kun Chen
Assistant Professor
Department of Statistics
University of Connecticut
Storrs, CT
November 18, 2013
Outline I
1
Chapter 6 Function of Random Variables
6.3 The Method of Distribution Functions
6.
Homework 8
Solutions!
1. Let X (, ). Derive the variance of X without using the Moment Generating
Function of X
Solution:
Proof.
V [X]
2
E[X ]
=
E[X 2 ] E 2 [X]
=
E[X 2 ] ()2
Z
=
Z 0
=
x2 fX (x)dx
x2 fX (x)dx +
Z
=
=
=
Z
0
x2 fX (x)dx
1
x
x1 e dx
()
0
STAT 3375Q: Introduction to Mathematical Statistics
Kun Chen
Assistant Professor
Department of Statistics
University of Connecticut
Storrs, CT
October 24, 2013
Outline I
1
Chapter 5: Multivariate Probability Distributions
5.2 Bivariate and Multivariate Pr
3.1. P(Y = 0) = P(no impurities) = 0.2
P A B 1 P Y 0 0.8 ,
so P Y 2 P A B P A P B P A B 0.1
P(Y = 1) = P(exactly one impurity) = P A P B 2P A B 0.7
3.9
The random variable Y takes on vales 0, 1, 2, and 3.
a. Let E denote an error on a single entry and let
Solutions for HW3
2.85(10) 2.89(10)
2.135(20)
2.91(10)
2.95(12)
2.110(10)
2.129(10)
2.131(8)
2.133(10)
2.89
a. 0, since they could be disjoint.
b. the smaller of P(A) and P(B).
2.91
If A and B are M.E., P( A B) = P(A) + P(B). This value is greater than 1
2.37
a. There are 6! = 720 possible itineraries.
b. In the 720 orderings, exactly 360 have Denver before San Francisco and 360 have San
Francisco before Denver. So, the probability is .5.
2.38 By the mn rule, 4(3)(4)(5) = 240.
2.42 There are three differe
Homework 5
Solutions!
1. Let X P oisson() where > 0.
Derive the variance of X (i.e. show that V [X] = )
Solution:
V [X] = E[X 2 ] E 2 [X]
also,
E[(X 1)X] = E[X 2 ] E[X]
E[X 2 ] = E[(X 1)X] + E[X]
= E[(X 1)X] +
E[(X 1)X] =
=
=
=
X
xS
X
(x 1)xpX (x)
(x 1)
Homework 6
Due: 10/14/2014
1. Problem 4.25 from the book (p 172)
Solution:
First we must determine the PDF of Y
fY (y) =
=
d
F (y)
dy Y
d
y0
dy 0,
d y, 0 < y < 0
dy 8
d y2
dy 16 ,
d
dy 1,
0,
1
8,
=
y
,
8
01,
2y<4
y4
y0
0<y<0
2y<4
y4
So, That means that
Quiz 10
NAME:
1. (15 pts) Let X and Y be continuous random variables with the following joint
PDF:
1
y 0<x<y <1
f (x, y) =
0 else
a) Find P (X + Y < 1)
Solution:
ZZ
f (x, y)dxdy
P (X + Y < 1) =
Z
x+y<1
.5 Z y
=
y=0 x=0
Z .5 Z y
=
=
=
=
Z
1
Z
1y
f (x, y)d
Quiz 8
NAME:
1. (10 pts) Evaluate the following integral:
Z 2
x x3
e 2 dx
2
0
Solution:
Z 2
x x3
e 2 dx
1
3
Z
x2 x3
3 e 2 dx
=
2
2
0
Integration by subsittution
x3
3x2
u=
du =
dx
2
2
u Ranges from 0 to
When x ranges from 0 to
Z
Z
2
3
1
x x
1 u
2
3 e
Quiz 9
NAME:
1. (10 pts) Let X Beta(,) have the pdf
(
fX (x) =
(+) 1
(1
()() x
0
x)1 if
0<x<1
else
where 0 < < and 0 < <
a) Let r be a real valued constant where r > . Prove that
E[X r ] =
( + r)( + )
()( + + r)
Solution:
Proof.
E[X r ] =
=
=
=
=
Z 1
(
Quiz 7
NAME: SOLUTIONS!
1. (10 pts) Let R U (0, 1). Suppose R was the radius of a random circle, with Area
equal to A = R2 . Find the mean and variance of the area of the random circle.
(i.e. Find E[A] and V [A])
Solution:
V [A] = V [R2 ]
= ( 2 )V [R2 ]
E
Quiz 5
NAME:
1. (10 pts) Suppose that you like to spend your Saturdays at the beach with a metal
detector looking for gold. Lets say that 1 in 10 metal objects buried in the sand
is actually a piece of gold. Let X be the number of things you will have dug
Quiz 2
NAME: SOLUTIONS!
1. (10 pts) Let A and B both be subsets of the superset S. Prove that (A B)C =
AC B C
Solution:
To prove that these sets are equal we must show that
a) (A B)C AC B C
b) AC B C (A B)C
Proof. a) We want to show that (A B)C AC B C
Let
Quiz 4
NAME: Solutions!
1. (10 pts) Let X be a random variable such that E[X] and V [X] exist and are finite,
and let a, b both be real valued, non-zero constants. Suppose that we define Y
to be the random variable such that Y = aX + b (this might happen
Quiz 6
NAME: SOLUTIONS!
1. (10 pts) Let X P ois() and let pX (x) be the pdf of X
a) Show that the ratio of successive probabilities satisfies
pX (x)
= for x = 1, 2, 3, .
pX (x 1)
x
b) For which values of x is pX (x) > pX (x 1)?
Solution:
a) Note, the rati
Quiz 3
NAME:
1. (10 pts) Let A, B, and C be events in the sample Space S with P (C) > 0. Prove
that the general additive rule of probability also applies to conditional probability.
In other words, prove that
P (A B|C) = P (A|C) + P (B|C) P (A B|C)
Soluti
Homework 9
Due: 11/4/2014
1. Problem 5.3 from the book (p 232)
Solution:
2. Problem 5.9 from the book (p 233)
Solution:
3. Problem 5.21 from the book (p 244)
Solution:
1
4. Problem 5.27 from the book (p 244)
Solution:
2
Challenge Question:
Let X1 and X2 b
Homework 7
SOLUITONS!
1. Evaluate
Solution:
R
(x
2
+ 1)2 ex dx
Z
2 x2
(x + 1) e
Z
dx
=
2
(x2 + 2x + 1)ex dx
Z
=
2 x2
x e
Z
=
Z
2xe
dx +
x2
Z
dx +
2 x2
x e
2
Z
e
dx + 0 + 2
x2
dx
0
0
2
Since 2xex is an odd function
Integration by Substitution
u=x
Homework 1
SOLUTIONS!
1. Problem 2.6 from the book (p 26)
Solution:
2. Let A and B be events from subset S. Prove that
P (A) = P (A B) + P (A B C )
Solution:
Proof. First, we will show that A = (A B) (A B C ).
A = A S Since A S
= A (B B C ) By the definit
1
Set Theory Definitions
A set is simply a collection of objects:
The set of counting numbers
The set of students in this classroom
The set of professors in the statistics department
Typical notation: A = cfw_. . .
Examples:
N = cfw_0, 1, 2, . . .
This lecture will focus on the law of total probability and Bayes rule, but first we
must cover some concepts.
1 Preliminaries
First, consider the following situation:
Let A, B1 , B2 , ., Bn be events of the sample space S such that
1. S = B1 B2 . Bn
2.
STAT 3375
Calculus Review
1 Taylor Series
1.1 Denition
Let f be a function which is dened and innitely dierentiable at all points on some interval (c, d) and let
a be some number in this interval. If f is analytic on this interval, then it can be shown th
1
Probability Experiments
Experiments
A probability experiement is an activity that involves chance that
leads to results, that can be repeated
Sample Points
A sample point is a possible result of a single trial, or repetition, of
an experiment
Event