Stat 241/541 Final Exam
12/17/2012, 2:00-4:30PM. This is a two-hour-and-a-half exam for which you
have three hours.
Important: You may choose 5 out of the following 6 problems to
work on. Please tell us what 5 problems to be graded!
1. (20 points) A docto
Week 7
Important Discrete Distributions
Expectation and Variance
Lecture 17. Bernoulli and Binomial Distributions. Expectation
Revisited.
Discrete Uniform Distribution.
We have seen some examples that all outcomes of an experiment are equally
likely. Let
Week 4
Lecture 8. Discrete conditional distribution.
Examples 1: A doctor gives a patient a test for a particular cancer. Before
the results of the test, the only evidence the doctor has to go on is that 1 woman
in 1000 has this cancer. Experience has sho
Week 6
Lecture 14
Random Walks
Drunkard Walk. Imagine now a drunkard walking randomly in an ideals
ized 1 dimensional city ( or 2 dimensional, or 3 and higher dimensional city).
The city is eectively innite and arranged in a 1 dimensional equally-spaced
g
Week 3
Lecture 5. Expectation
Probability Density Function: Let f (x) 0 and
P (E ) as following
Z
P (X 2 E ) =
f (x) dx.
R
f (x) dx = 1. Dene
E
Are the probability axioms satised?
It is important to observe that there a similar paradox in the calculus
Za
Week 2
Lecture 3. Expectation and Probability axioms.
Random variable. A random variable is a real-valued function dened on
the sample space, i.e., X (! ) is a function from to R. For example, for =
fBB; BG; GB; GGg, your X could be the number of boys, th
Midterm Exam, 2012 Fall (Time: 9:00-10:15am, Oct. 19)
1. Let U1 ; U2 ; : : : ; Un be i.i.d. from U [0; 1], each with density function
f (x) =
1
0
0<x<1
.
otherwise
Let 0 < a < b < 1. Dene
Xi =
1 if 0 < Ui < a
, and Yi =
0
otherwise
1 if 0 < Ui < b
.
0
oth
Homework 2
Due September 13.
Chapter 2.2: 2, 4, 6, 8, 12.
Problem 6: Let X
U (0; 1).
(i) Find the density of Y = 1=X and EY .
X
(ii) Find the density of Y = tan 2 and EY .
1
STAT 241/541, Probability Theory with Applications
Fall 2013
Instructor:
Harrison H. Zhou (huibin.zhou@yale.edu)
O ce hours: Wednesday 4:00-6:00pm (tentative) or by appointments, Room
204, 24 Hillhouse Ave., James Dwight Dana House.
T.A.:
Corey Brier <cor