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
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
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: pldavies@ucdavis.edu Chris Dienes 1117 Mathematical Scie
STA131A: Homework 7
Prof. James Sharpnack
Due 6/1 in class
1. 7.P19
Solution:
(a) This is a Geometric random variable minus 1 (because the number of insects before the first
type 1 is the first type 1 time minus 1), so the mean is P11 1. It is not a big d
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
HANDOUT: Review of Some Results from Calculus
For more details, please consult a Calculus text.
1. Series
(a)1+a+.+a =1Gm ifayé 1[since (1a)(1+a+. +a") = 1 a].
lva.
(b) 1 + a+a2+ = 1:26 =JL1110131": = if la] < 1;
[follows from (23.)
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
Name
Student ID
STA 131A  MIDTERM 1
SPRING 2016 / APRIL 18, 2016
Problem 1 (24 points)
An urn contains 2 blue balls (B), 1 red ball (R) and 1 white ball (W). The balls are identical except for
color.
The following random experiment is performed: The 4 ba
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
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
(
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 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 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
/STATISTICSI31A HANDOUTContinuous Distributions
1.4
Failure me
Failure um. THE BETA DISTRIBUTION
Denition oI the Beta Distribution
It is said that a random variable X has a beta distribution with parameters a and
5 (a > o and p > 0) ifX has
STAT 131
HANDOUT JOINT DISTRIBUTIONS
Discrete Joint Distributions
Suppose that a given experiment involves two random variables X and Y, caCh of
which has a discrete distribution. For example, if a sample of theater patrons is
selected, one random variabl
1
STATISTICS 131A Sample Midterm
1. A class has 50 students; among them 20 have iPhones, 12 have Samsung, 6 have BlackBerry, 8 have other cell phones, and 4 students do not have cell phones. If a random sample
of ve students are chosen, what is the probab
STATISTICS 131A
PROBABILITY THEORY
Fall 2015
MIDTERM
Student ID number
Print name
Sign name
Score:
1:
2:
3:
Total:
2
1. (16 points) Jim and his daughter, Sarah, choose who will mow the lawn by playing a chance
game.
a. Jim has three marbles in his pocket;
STA131A: Homework 2 Solutions (do not distribute)
Prof. James Sharpnack
Due 4/11 in class
1. 2.P24
Solution: Let x,y be the outcome of each roll
P cfw_x+y = k =
6
X
P cfw_x+y = kx = iP cfw_x = i =
i=1
6
X
P cfw_y = kiP cfw_x = i =
i=1
1
(mincfw_(k 1), 6
STA131A: Homework 6
Prof. James Sharpnack
Due 5/16 in class
1. 6.P44
Solution: We can compute this from the following: cfw_X(3) > X(1) +X(2) = cfw_X3 > X1 +X2 cfw_X1 >
X2 + X3 cfw_X2 > X1 + X3 and each of these events is mutually exclusive and they hav
STA131A Midterm 2: 5/20
Your Name:
Instructions: You may leave your answers as fractions and even as factorials (e.g. 4!). Show all your work,
points may be deducted for no justification of an answer. You should evaluate any integrals by yourself.
1. Let
STA131A: Homework 3
Prof. James Sharpnack
Due 4/18 in class
1. Three cooks A,B,C, bake a special cake and with probabilities 0.02, 0.03, 0.05 respectively the cake fails
to rise. In the restaurant where they work, A bakes this cake 50% of the time, B bake
STA131A: Homework 5
Prof. James Sharpnack
Due 5/9 in class
1. 5.P37
Solution:
(a) P cfw_X > 1/2 = 1 P cfw_1/2 < X < 1/2 = 1/2
(b) P cfw_X < x = P cfw_x < X < x = 2x/2 = x, 0 x 1 hence we can immediately see that it is
uniform(0, 1). One way to see thi