395 practice exam.
Here is a short normal table:
x
0
0.25 0.52 0.84
(x) 0.5
0.6
0.7
0.8
1.28
0.9
1.64
0.95
1.96 2.05
0.975 0.98
2.33
0.99
2.58
0.995
1. Membership of the Mensa society is open to persons who have attained a score within
the upper two perce
Chapter 5
Chapter 6
Chapter 7
Inequalities
Markov Inequality*
If a random variable X can only take nonnegative values, then
P(X a)
E (X )
, for all a > 0
a
Chebyshev Inequality*
If X is a random variable with mean and variance 2 , then
P(|X | c)
2
, for
Homework #4 Key
Problem A
1. Let state s = 1. The problem is asking you to calculate the mean rst passage time t3 :
t1 = 0
3
t2 = 1 +
p2j tj
j=1
0.3t2
= 1 + p22 t2 + p23 t3
= 1 + 0.70t2 + 0.10t3
= 1 + 0.10t3
3
t3 = 1 +
p3j tj
j=1
=
=
0.1t3 =
0.3t2 1 =
STAT/MATH 395A
Problem Session 1. Find the group with the circled number (map on whiteboard). Start
working together on the circled question. Continue with the next question (1 is the next
question after 8). Do as many as you have time for. Question 8 is
STAT/MATH 395
Sample Midterm Exam
Do as many questions as you can. Each question is worth 20 points. You are allowed one
sheet of notes. A normal table is attached if you need it.
1. Let X be a continuous random variable with density f(x) = k x-3, where k
5.2!
Temperature!
S395!
A certain chemical reaction achieves a temperature,
X, varying from experiment to experiment according
to the pdf! fX (x) = x e -x2/2 , x 0!
5.1!
S395!
Chapter 5:
Functions of
random variables!
where X is measured in degrees Celsiu
STAT/MATH 394, Summer 2010 A
Hoyt Koepke
hoytak@stat.washington.edu
June 20, 2010
STAT/MATH 394, Probability I, covers the basic elements of probability theory. The goal is
to provide a solid grounding in understanding and working with practical problems
STAT/MATH 395
Case study
The setting of air quality standards for ozone
The Environmental Protection Agency (EPA) is charged with setting stardards for
air and water quality in the United States. In the summer of 1997 a revised
standard for ozone (and for
4.1!
S395!
4.2!
S395!
STAT/MATH 395!
Peter Guttorp
!
peter@stat.washington.edu
!
4.3!
S395!
Chapter 4
Expected Values!
Expected value!
Variance!
Covariance!
Law of large numbers!
Class structure!
Homework due in class on Fridays (20% of grade)!
!
Midterm
S395!
1!
S395!
2!
The Scottish chest
measurements!
109
VOL. 79, NO. 2, APRIL 2006
Chapter 6
The normal distribution!
Relative
frequency
0.2
0.15
Some history!
The central limit theorem!
Normal approximations!
0.1
0.05
33
35
37
Figure 8
39
41
43
45
47
Che
Homework #3 Key
Problem A
1. This problem is asking you to compute the two-step transition probability r33 (2). This
can be computed from the Chapman-Kolmogorov equation:
3
r33 (2) =
r3k (1)pk3
k=1
=
=
=
=
p31 p13 + p32 p23 + p33 p33
0.01 0.05 + 0.09 0.10
Homework #1 Key
If X P oisson( = 2), then the distribution of X has mean = 2 and variance 2 = 2.
1. Markov Inequality:
E(X)
by Markov Inequality
5
2
=
= 0.4
5
P (X 5)
Chebyshev Inequality:
P (X 5) = P (|X 2| 3)
2
by Chebyshev Inequality
9
2
=
0.22
9
2.
STAT/MATH 394A W14
Final exam
1. If P(A|B) = P(B|A), are A and B independent? Explain your reasoning.
2. At one time the U.S. Senate Committee on Labor and Public Welfare was
investigating the feasibility of
Gamblers Ruin Problems
Slide 15
Let $0 be the state 0, $100 be the state 1, $200 be the state 2, $300 be the state 3, and
$400 be the state 0. State 2 is the initial state, states 0 and 4 are absorbing states, and states
1, 2, and 3 are transient states.
Real Life example
CLT Activity
Chapter 5: Central Limit Theorem (CLT) Activity
MATH/STAT 395: Probability II
Roddy Theobald
Summer 2014
1 / 16
Real Life example
CLT Activity
Real Life Example (Cobb and Gehlbach, 2004)
Kristen Gilbert was a nurse at the VA
Von Neumann Extractor
Random Incidence Paradox
Splitting and Merging
Challenge Problems
Chapter 6: Additional Topics in Bernoulli and
Poisson Processes
MATH/STAT 395: Probability II
Roddy Theobald
Summer 2014
1 / 16
Von Neumann Extractor
Random Incidence
Section 7.1
Section 7.2
Section 7.3
Homework
Chapter 7, Part 1: Discrete-Time Markov Chains
MATH/STAT 395: Probability II
Roddy Theobald
Summer 2014
1 / 42
Section 7.1
Section 7.2
Section 7.3
Homework
Stochastic Processes
Each of these sequences of random
1. Andrea has found that she wins about 28% of the time when playing a solitaire named
Agnes Bernauer on her computer.
(a; 5 points) Calculate the expected number of games before the next win?
(b; 5 points) What is the probability that she will get her se
STAT 395, Sp14.
Some geometric sums (pp.576-577 in the book).
1
a k = 1 a for a < 1, since (1 a) a k = (a k a k +1 ) = 1 a + a a 2 + a 2 .
k=0
k=0
k=0
aL
ak =
since a k = a L a k L = a L a j
1 a
k=L
k=L
k=L
j=0
d k
d k d $