Math 302, assignment 8 solutions
1. a. Let X1 = Geom(p1 ) and X2 = Geom(p2 ) be independent Geometric random variables. What is the
distribution of X1 + X2 ? (compute either the c.d.f. or the p.m.f., but state which. Justify your answer.
b. Repeat this fo
Math 302. Assignment 8
Due Mar. 16
Exercises from the textbook (Grinstead & Snell 2nd revised edition)
1. Section 2.2 (p.72) Ex. 6
2. If X is an exponential random variable with parameter , and c > 0, show that
cX is exponential with parameter /c.
3. If X
Math 302, assignment 7
Due Nov. 2
Note: A few questions on WebWork.
1. If X is a N (0, 1) r.v., find the p.d.f. of X 2 . (Hint: use the CDF).
p
1/
|X|. (Hint: same as for 1.)
2. If X is a Cauchy r.v. (density 1+x
2 ), find the p.d.f. of
3. a. Prove that i
Math 302, assignment 11
Do not hand in.
1. If X is the sum of 100 six sided dice, what does Chebyshevs inequality say about P(X < 300)? What
is the CLT approximation?
2. Suppose each customer at a store pays on average $12, and the variance is 52 . If the
Math 302, assignment 10
Due Nov. 23
Note: there are questions on WebWork as well.
The correlation of variables X, Y is defined by
(X, Y ) =
Cov(X, Y )
.
Var X Var Y
It measures how linearly dependent variables are.
1. Show that if Y = aX + b for some fix
Math 302, assignment 10
The conditional expectation E(X|Y ) is defined as
continuous case.
Due Nov. 30
P
xp(x|y) in the discret case and
R
xf (x|y)dx in the
1. X, Y have joint density 21 yeyx/y for x, y > 0 and 0 if either is negative.
a. Find the margina
Math 302 Sample exam
Instructions
Explain your reasoning thoroughly, and justify all answers (even if
the question does not specifically say so).
No aids are permitted (calculators,notes,etc).
A formula sheet is provided (see piazza and online).
Questi
Math 302, assignment 5
Due Oct. 19
Note: the last WebWork assignment was extended to this week.
1. A random variable has density f (x) = cx3 on [2, 5]. Find c, find the CDF F (t) and find P(X 4).
2. A dice is thrown. If the result is Y , then we let X be
Math 302, assignment 6
Due Oct. 26
1. Suppose
R X is a continuous random variable and X 0 always. Let F be its CDF. Show that
EX = 0 1 F (t)dt. (Hint: use integration by parts.)
Bonus: Show this also for discrete positive random variables.
2. a. For the f
Math 302, assignment 8
Due Nov. 18
Note: A few questions on WebWork.
1. A dartboard centered at the origin has radius 6. Let (X, Y ) be the random location of a dart thrown
by a competent player. Assume X and Y have joint probability density function
(
p
Math 302, assignment 4
Due Oct. 12
Note: there are several questions on WebWork for this week as well.
1. An experiment is repeated, and the first success occurs on the 8th attempt. What is the success
probaility for which this is most likely to happen? (
Concentration camp kids study:develop peer attachment instead of caregiver attachment
All the same age kids involved in this study
Harlows infant monkey study: what we saw in 331
Wire mom vs cloth mom(food vs love)
Wire means: just only equipped with fake
MATH 302
b
ASSIGNMENT
T S
b
Q [ points]. (a) How many dierent ways are there to share identical
candies among distinguishable children?
(b) How many dierent ways are there to share the candies if each child is to receive at
least one candy?
Q [ points].
Math 302 Sample midterm I
Instructions
Explain your reasoning thoroughly, and justify all answers (even if the question does
not specifically say so).
No aids are permitted (calculators,notes,etc).
Questions
1. (a) Define carefully what it means for P t
1. Define various terms (p.m.f., p.d.f., c.d.f., expectation, variance, independent variables,
etc)
Solution: See definitions in the notes.
2. Suppose a random variable X has density function f (x) = aex on [0, 1] and 0 for x < 0
and for x > 1.
(a) Find a
Math 302 Sample midterm II
Instructions
Explain your reasoning thoroughly, and justify all answers (even if the question does
not specifically say so).
No aids are permitted (calculators,notes,etc).
A formula sheet is provided (see piazza and online).
Math 302 Sample midterm I
Instructions
Explain your reasoning thoroughly, and justify all answers (even if the question does
not specifically say so).
No aids are permitted (calculators,notes,etc).
Questions
1. (a) Define carefully what it means for P t
Math 302, assignment 5
Due Oct. 19
Note: the last WebWork assignment was extended to this week.
1. A random variable has density f (x) = cx3 on [2, 5]. Find c, find the CDF F (t) and find P(X 4).
R5
solution. We have 2 cx3 dx = 1, which gives c(54
Rt
CDF
1
c Anthony Peirce.
Introductory lecture notes on Partial Differential Equations -
Not to be copied, used, or revised without explicit written permission from the copyright owner.
Lecture 13: Full Range Fourier Series
(Compiled 3 March 2014)
In this lect
c Anthony Peirce.
Introductory lecture notes on Partial Differential Equations -
Not to be copied, used, or revised without explicit written permission from the copyright owner.
1
Lecture 26: Circular domains
(Compiled 3 March 2014)
In this lecture we co
Math 302. Assignment 4
Due Feb. 10
Exercises from the textbook (Grinstead & Snell 2nd revised edition)
1. Section 4.1 (p.154) Ex. 32
Note: we assume that the passable/non-passable status of different roads are independent.
2. Section 4.1 Ex. 43
hint: the
Math 302. Assignment 1
Due Jan. 20
Exercises from the textbook (Grinstead & Snell 2nd revised edition)
1. Section 1.2 (p.35) Ex. 6
Hint: there is a certain C > 0 (to be determined) such that the distribution function
is of the form m(i) = C i for i = cfw_
Math 302. Assignment 3
Due Feb. 3
Exercises from the textbook (Grinstead & Snell 2nd revised edition)
1. Section 3.2 (p.115) Ex. 20
2. Multinomial coefficients
For m 1 and n1 , . . . , nm positive integers, with the notation n for the sum
n1 + + nm , deno
Top Ten Summation Formulas
Name
1.
Binomial theorem
Binomial series
Summation formula
!
n
X
n
xnk y k
(x + y)n =
k
k=0
!
k
2.
Geometric sum
X
integer n 0.
ar k = a
ar k =
k=0
3.
Telescoping sum
|x| < 1 if 6=
n
k=0
Geometric series
integer n 0
k
x = (1 +
Problems from Chapter 3 Handout:
13. A die is rolled three times. What is the probability that you get a larger number
each time?
Since only one permutation will be strictly increasing of the 6 possible from
MATH 302 INTRODUCTION TO PROBABILITY SUPPLEMENT
Elementary calculus and some special functions
In this supplement we discuss some important mathematical concepts that you need to
know.
Differentiation and integration.
Principle 1 [Differentiating a polyno
July 25, 2014
Math 302 HW assignment 4. Due Friday, August 1 at start of class
Note: Write your student number on each page that you submit. Show all your work! Separate the
solutions of different exercises with a line. Draw a frame around your final answ
MATH 302
MIDTERM #1
Instructor: Dr. T. Hulshof
1 Last name:
First name:
Student number:
Instructions
Write your answers legibly below each question. The space provided for each question should be sufficient, but if you do run out of space you may write y