STA 131A
Midterm I
October 20, 2006
Note: Show all your work, in particular the details how you a.rrive at the answers. Provide the
formulas but do not work out the final inumericad values.
1. (25 points) There are 6 students to be assigned raadomly t o f
Name
Final Exam
STA 131A
December 11, 2006
Note: Show all your work, in particular the details how you arrive a t the answers. Provide th.e
final numerical values unless you are instructed N O T t o do so.
1. (42 ~ o i n t s )Let X and
'Ybe random va'riab
FALL QUARTER 2010 Statistics 131A: Probability Theory
Division of Statistics
Instructor:
Professor Laurie Davies Office hours: M,W F 1.10 2.00pm 4101 Mathematical Science Building Phone: ? ? email: [email protected] Chris Dienes 1117 Mathematical Scie
2.12 For any three events A, B, and C in a sample space S, Show the transitive
property relative to E holds (ie, A g B and B E C implyr that A E C ).
2.14 Establish the identity:
UjAj =A1U(AfA2)U(A1AgA3)Un
Hint. As in Exercise 2.13. 3.4 The 12v. X has
STA 131A
Probability Theory
Spring 2015
Discussion 9
Problem 1
Show that X and Y are identically distributed and not necessarily independent, then
Cov(X + Y, X Y ) = 0
Ans:
Cov(X + Y, X Y ) = Cov(X, X) + Cov(X, Y ) + Cov(Y, X) + Cov(Y, Y )
= V ar(X) Cov(X
12.
15.
me I different} =P{6, dierent}!P{different}
= P{lst = 6,2nd :6 6} + .P{ist at 6,2nd = 6}
SM
n 2 U6 5l6 cm
5.6
0 have been solved by using reduced sample spacefor given that outcomes differ it
could als
ing for the probability that 6 is chOSen when
STA 131A
Probability Theory
Spring 2015
Discussion 8
Problem 1
The random vector (X, Y ) is said to be uniformly distributed over a region R in the plane
if, for some constant c, its joint density is
c, if (x, y) R
0, otherwise
f (x, y) =
1. Show that 1/c
STA 131A
Probability Theory
Spring 2015
Discussion 6
Problem 1
An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white
and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4
ball
STA 131A
Probability Theory
Spring 2015
Discussion 5
Problem 1
Let X be a binomial random variable with parameters n and p. Show that
E
1 (1 p)n+1
1
=
X +1
(n + 1)p
Ans:
1
) =
E(
X +1
n
i=0
n
=
i=0
=
=
1
i+1
n
i
pi (1 p)ni
n!
pi (1 p)ni
(n i)!(i + 1)!
1
(
1. In the book and in class notes we showed using the relative frequency denition of probability,
that if events A and B are disjoint, that is PM. F] B = (a, then P(A U B] = Pcfw_A + P[B]I.
Use the relative frequency denition of probability to show that f
1. A continuous random variable Y with support (0, 1) and pdf
I(or + cfw_3)
aim!
is called a beta distribution. Find E(Y) Use the method (explained above) of doing
the integration by recognizing the kernel of the beta distribution when you do the
integrat
1
STA131A spring 2017 Homework 1
Due at the beginning of class on Wednesday April 12, 2017
Questions relate to the material in chapter 2.
PLEASE FOLLOW ALL INSTRUCTIONS.
These instructions are for your protection to eliminate the consequences of your home
1
STA131A spring 2017 Homework 2
Due at the beginning of class on Wednesday April 19, 2017
Questions relate to the material in chapter 3.
PLEASE FOLLOW ALL INSTRUCTIONS.
These instructions are for your protection to eliminate the consequences of your home
1. Using the Bernoulli variables X1 and X2 dened above for A1 and Ag, how would you write
P(A1 LJ Ag) in terms of a probability statement for X1 and X2? 2. Let A, B, and C be events. Dene three Bernoulli variables as
ifA 1 ifB 1 if C
X: Y: Z:
0 ifA 0 ifB
1. Tenants in a large apartment complex are allotted to sum cfw_legs1 cats, and rabbits, but are
not allowed to own more than two pets. Write down the sample space for the pets owned by
the tenants in a randomly selected apartment. Use the following notat
1. Let X be a discrete random variable with support cfw_1I . . . ,n and density function
m3) = P(X = 3)
l"
for$=l,.,n, (1)
where n, is an integer. The distribution of a random variable 1with this density function
is called a uniform discrete distribution.
1.4 Suppose that the probability that both of a pair of twins are boys is 0.30 and
that the probability that the)r are both girls is 0.26. Given that the probability
of the rst child being a b0)r is 0.52, what is the probability that:
(i) The second twin
STA 131A
Probability Theory
Spring 2015
Discussion 7
Problem 1
The lifetime in hours of an electronic tube is a random variable having a probability density
function given by
f (x) = xex x 0
Compute the expected lifetime of such a tube.
Ans:
xxex dx
E(X)
STA 131A
Probability Theory
Spring 2015
Discussion 4
Problem 1
Five distinct numbers are randomly distributed to player numbered 1 through 5. Whenever
two players compare their numbers, the one with the highter one is declared the winner.
Initially, playe
Statistics 131A HANDOUT # 2

Mean & Variance

Compute the mean and variance of Geometric random variable.
Let X Geom(p).
where q = 1  p.
(a). To find E ( X ) , there are two methods.
Method 1:
Method 2: Let
Hence A =
1
 1
P
=+E(X)=PA==(1d2
p2
p2
1
Statistics 131A HANDOUT
#3

Review of Chapter 4
A batch of N products consists of m defective items and N  m good (nondefective) items. You
will sample from this batch either w i t h o r w i t h o u t replacemen~tand are interested in several
random va
Statistics 131A
Solutions of Homework 1
Fall 2015
Due: Friday, 10/2/2015
All problems are from the textbook Introduction to probability.
1. Problem 1 from section 1.9
Ans: In the word MISSISSIPPI there are 11 letters: 1 M, 4 Is, 4 Ss, and 2 Ps. Hence ther
Statistics 131A
Solutions of Homework 3
Fall 2015
Problem 7 (a) Let, D= the event that there is a double headed coin.
H= the event that a chosen coin gives 7 heads in 7 tosses.
A= the event that the chosen coin is double headed.
P (DH) =
=
=
P (HD)P (D)