Chapter 4
Mathematical Expectation
4.1 & 4.2 Mean of a Random Variable
The value you expect to get in a statistical experiment is called the
Mean/Expectation/Expected Value of the random variable,
denoted by E[X ].
Example 1
Tossing a coin once
Let X deno
Review
Section 2.7: Independent Events
Example 8: When you flip a thumbtack, it can land
either point up (with probability 0.32) or point down
(with probability 0.68). When we flip five times,
what is the chance that it lands point down on all
five flips?
Stat 4201 Introduction to Mathematical Statistics I
Autumn Semester 2012
Lecture: MWF 9:10-10:05 AM in Evans Lab 1008
Instructor: Dennis Pearl
Office: 440F Cockins Hall
Phone: 292-3887
Office Hours: TW 10:20 am or by appointment
Email: [email protected]
TA:
Geometric Distributions and Negative Binomial
Geometric Distribution
We repeat independent Bernoulli trials with success probability p.
Let X be the number of trials until the first success occurs. The
distribution of X is
P(X = x) = p(1 p)x1 ,
x = 1, 2,
Introduction to Mathematical Statistics
Lecture note 3
Instructor: Yuan Zhang
3.3 & 3.4 Continuous Probability Distributions
Example 1
Suppose we received a report of an accident on a freeway of 200
miles long. We are interested in how likely that it occu
Introduction to Mathematical Statistics
Lecture note 2
Instructor: Yuan Zhang
2.8 Total Probability Theorem and Bayes Theorem
Total probability theorem
We want to calculate P(A), but it is sometimes easier to find
P(A|B1 ), . . . , P(A|Bm ) and P(B1 ), .
Chapter 6
Some Continuous Probability Distributions
Name
Uniform
Beta
Notation
U[a, b]
Beta(, )
Normal
N (, 2 )
Exponential
Gamma
Chi-Squared
Exp()
(, )
2
= 2 , 2
P.D.F. f (x)
a x b.
x)1 , 0 < x < 1
1
ba ,
1
1
B(, ) x
(1
1
2
2
,
expcfw_ (x)
2 2
e x ,
<
4.8 Conditional Expectations
Definition: Conditional Expectation
If X is a discrete random variable, the conditional expectation of
u(X ) give Y = y is
X
E u(X )|Y = y =
u(x)P(X = x|Y = y )
x
Correspondingly, if X is a continuous random variable, the
con
Stat 4201, Spring 2017: HW 6 Solutions
In the 8th edition of the textbook do exercises: 5.16, 5.17, 5.23, 5.24, 5.40, 5.41, 5.51
In the 7th edition, these are: 5.16, 5.17, 5.23, 5.24, 5.40, 5.41, 5.51
5.16 If X Negative Binomial(k, ), then, in the books n
Stat 4201, Spring 2017: HW 5 Solutions
In the 8th edition of the textbook do exercises: 3.74, [with reference to 3.42 and 3.70, find Cov(X, Y )],
4.47, 4.49, 4.50, 4.58, 4.64, 4.80
In the 7th edition, these are: 3.74, 4.42, 4.46, 4.48, 4.49, 4.57, 4.62, 4
Stat 4201, Spring 2017: HW 4 Solutions
Problem 1. In the 8th edition of the book, do problems: 4.8, 4.10, 4.20, 4.33, 4.37, 4.38
In the 7th edition of the book, these are: 4.8, 4.10, 4.20, 4.34, 4.37, 4.38
4.8
Z
E(X)
3 1
=
1
Z
x2 dx +
x f (x)dx =
=
Z
0
3
STAT 4201, Spring 2017: Homework 7
This homework is due on Wednesday March 8 and will be graded by Guowei Li. Submit this
homework to the TA at the recitation you registered. Do not submit the homework
to the professor. You are encouraged to discuss with
Section 3.5: Multivariate Distributions
Example 4: A small bed-and-breakfast has 3
rooms for rent including two with king-sized beds
and one with two queen-sized beds. The
probability distribution for the number of rooms, X,
rented on a weekend is given b
Practice Midterm
1. A book has four typos. After each rereading, an
uncorrected typo is corrected with probability 0.6. Each
correction of different typos is independent, and each of
the re-readings is also independent one from another.
(a)What is the pro
Review
Section 4.6: Product Moments
With two or more variables there are other moments
to consider for the joint distribution. For example, for
two variables X and Y, the jth and kth product moment
about the origin is defined by
'
j ,k
jk
= E ( X Y ).
Wh
Review Section 3.1: Random Variables
Example 2: Hurricane/Storm categories take each
value of wind speed from the sample space and
assign it a number.
This is an example of a random variable X
(definition: a real-valued function defined on the
elements of