MATH/STAT 418 Section 002 2008/06/26 Exam 1 Sample
Instructions: 1. There are six problems in this exam. The entire exam is worth 100 points and the point distribution is noted next to e
Stat 418 HW #4
Problem 4.6
1/8 if X = 3
3/8 if X = 1
P (X) =
3/8 if X = 1
1/8 if X = 3
Problem 4.7
(a) S = cfw_1, 2, 3, 4, 5, 6
(b) S = cfw_1, 2, 3, 4, 5, 6
(c) S = cfw_2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
Stat 418 HW #6
Problem 6.8
Ry
(a) fY (y) = c
(y 2 x2 )ey dx
y
= 43 cy 3 ey ,
Since
R
0<y<
fY (y)dy = 1, c =
0
R 1
(b) fX (x) =
|x|
8 (y
2
1
8
x2 )ey dy
= 14 e|x| (1 + |x|)
< x <
( by integration by
Stat 418 HW #3
Problem 3.7
It is the conditional probability of G given B
where G is the event that there is at least one girl, B is the event that there is
at least one boy
P (G|B) =
1/2
2
P (GB)
=
=
Exam 2 Solutions
STAT/MATH 418
Spring 2017
1. As the mean of the exponential is given to be 5 minutes, the rate parameter = 1/5. The lack of
memory property of the exponential thus gives the result e4
Notes 21
Desired outcomes from last class
Students will be able to:
Notes 21
8
Limit Theorems
Markovs Inequality: For P(X 0) = 1, a > 0,
derive the moment generating function of a r.v.;
find
E(X r )
f
Notes 20
Desired outcomes from last class
Students will be able to:
derive mean of sum of random variables;
derive expected value of function of two random variables;
Notes 20
Zx
=
derive variance of
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engineering
Chapter 2, Yates & Goodman
CHAPTER 2
Discrete Random Variables
1
STAT/MATH 418: Intro to Probability and
Stochastic Process
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engineering
Chapter 1, Yates & Goodman
CHAPTER 1
Experiments, Models, and Probabilities
1
STAT/MATH 418: Intro to Probability and
Stoch
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engineering
Chapter 3, Yates & Goodman
CHAPTER 3
Continuous Random Variables
1
STAT/MATH 418: Intro to Probability and
Stochastic Proce
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engineering
Chapter 5, Yates & Goodman
CHAPTER 5
Random Vectors
1
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engi
STAT/MATH 418
Solutions to Exam 1
Spring 2017
5 5
10
5 5
5 5
1. (a)
. (b)
+
+
.
8
3 5
4 4
5 3
2. E = you pass at least 2 exams = either pass all the exams or pass exactly 2 exams. As
E = A1 A2 A3
Lecture 04
Desired outcomes from last class
exploit the multiplication principle when outcomes are equally likely;
Lecture 04
recognize n Pr and
situations; know formulas;
Conditional Probability
Deni
Lecture 05
Desired outcomes from last class
dene the conditional probability of A given B using a formula;
understand intuitively what P (A | B ) means;
solve conditional probability problems using fo
Lecture 07
Desired Outcomes from last class
dene countable, discrete, and support;
dene the p.m.f. of a discrete random variable;
evaluate an unknown multiplicative constant in a p.m.f.
Lecture 07
2.2
Lecture 06
Desired Outcomes from last class
Lecture 06
2.1
Random Variables
state two forms of the Law of Total Probability;
Random variable is a mapping X from S to R.
solve problems using Bayes Theo
Lecture 01
Announcements
Class attendance is important for this course
http:/www.stat.psu.edu/~babu/418/
http:/sites.stat.psu.edu/~babu/418/418SpringSy14.html
Lecture 01
1.1 - 1.2
Set Theory, Basic Co
Stat 418 HW #9
Problem 9.2
3
P (T > s) = e 60 s
Problem 9.7
(n)
By Theorem 2.1, for an ergodic Markov chain, j = lim Pij
n
j are the unique nonnegative solution of
M
P
k Pkj (1)
j =
exists, and the
k=
Stat 418 HW #2
Problem 3.
20!
Let ij be the job for a person j. Then the problem is same as assigning 1, , 20
to i1 , , i20 . It is a permutation of 1, , 20, i.e., 20!
Problem 8.
(a). Since there are
Stat 418 HW #5
Problem 5.2
R
xex/2 dx = 2xex/2 4ex/2
c
R
xex/2 dx = 1
implies
c = 1/4
0
P (X > 5) =
1
4
R
xex/2 dx
5
= 14 (10e5/2 + 4e5/2 )
=
14 5/2
4 e
Problem 5.3
No, in both cases, f (1) = C implie
Stat 418 HW #7
Problem 7.18
(
1 match on card i
1
, then E[Ii ] = 13
Let Ii =
0 o.w.
52
P
1
=4
E[number of matches] = E[ Ii ] = 52 13
i=1
Problem 7.31
E[X1 ] = 72
E[X12 ] = 61 (12 + 22 + 32 + 42 + 52
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engineering
Chapter 10, Yates & Goodman
CHAPTER 10
Stochastic Processes
1
STAT/MATH 418: Intro to Probability and
Stochastic Processes
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engineering
Chapter 6, Yates & Goodman
CHAPTER 6
Sums of Random Variables &
Moment Generating Functions
1
STAT/MATH 418: Intro to Proba
STAT/MATH 418: Intro to Probability and
Stochastic Processes for Engineering
Chapter 4, Yates & Goodman
CHAPTER 4
Pairs of Random Variables
1
STAT/MATH 418: Intro to Probability and
Stochastic Process
Chapter 2 Concept Quiz
1. Which of the following are examples of discrete random variables? (Select all that apply.)
a. Number of customers that arrive at a shop
b. Time spent by a customer in a shop
Chapter 4 Concept Quiz
1. Please rewrite the following in terms of constants, (), and ():
a. , (1, ) = (1)
b.
, (, ) = 1
c.
, (, 3) = 0
d.
, (, ) = 0
2. Which of the following are definitions of in
Chapter 10 Concept Quiz
1. True or false: An i.i.d random sequence is a discrete-time, discrete-value stochastic
process. False. It can be discrete- or continuous-valued.
2. Fill in the blanks for the