Lecture Notes for Introductory Probability
Janko Gravner
Mathematics Department
University of California
Davis, CA 95616
gravner@math.ucdavis.edu
January 5, 2014
These notes were started in January 2009 with help from Christopher Ng, a student in
Math 135
Math 135A January 25, 2016
POISSON APPROXIMATION TO BINOMIAL
El 5 _= If ORG Math 135A January 25, 2016
WHY?
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Cm - (CLCU KW a? rK a.) 245.
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OctW V Mtg .
El 5 _= If ORG Math 135A January 25, 2016
EXAMPLE
Om W A. MW SubsCKVQ.
Math 135A
S OME P ROBLEMS
Ex You roll three dice in a row. What is the probability of
having no repeated numbers?
January 22, 2016
Math 135A
S OME P ROBLEMS
Ex You roll three dice in a row. What is the probability of a
strictly increasing sequence?
Januar
Math 135A
E XAMPLE
Ex Length of longest subsequence
January 27, 2016
Math 135A
G EOMETRIC R ANDOM VARIABLES
January 27, 2016
Math 135A
E XAMPLE
January 27, 2016
Math 135A
C ONTINUOUS R ANDOM VARIABLES
January 27, 2016
Math 135A
C ONTINUOUS R ANDOM VARIABL
Math 135A
VARIANCE
January 25, 2016
Math 135A
January 25, 2016
U NIFORM R ANDOM VARIABLES
Takes on the values x1 , x2 , . . . , xn with probability
1
n
for each.
Math 135A
B ERNOULLI R ANDOM VARIABLES
January 25, 2016
Math 135A
B INOMIAL R ANDOM VARIABLES
Math 135A
February 3, 2016
N ORMAL R ANDOM VARIABLE
A random variable X is normal with parameters , if it has
probability density function
(x)2
1
f (x) = e 22
2
Math 135A
N ORMAL R ANDOM VARIABLE
February 3, 2016
Math 135A
N ORMAL R ANDOM VARIABLE
Februa
HW: ASSIGNMENT 3
Due: Tuesday, October 18th
Warm Up: You should also do the problems in Section 4 of the notes.
1. There are three bags: A (contains 2 white and 4 red balls), B (8 white, 4 red)
and C (1 white 3 red). You select one ball at random from eac
Math 135A, Winter 2010.
March 20, 2010.
FINAL EXAM
Instructions: Each of the 6 problems has equal worth. Read each question carefully and
answer it in the space provided. YOU MUST SHOW ALL YOUR WORK TO RECEIVE
FULL CREDIT. Calculators, books or notes are
Math 16A
Homework 1
Homework 2
This homework is suggested study material for quizzes and exams.
1. An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without
replacement, what is the probability that the rst 2 selected are w
Math 135A, Winter 2013.
Extra Credit Problem 1
Important notes. Deadline to turn in any work on this problem is the last day of classes, Mar. 18,
2012, noon. Only a well-presented, correct, and substantial work on this problem will receive any
credit. In
Math 135A Midterm 2 Winter 2014
NAME(print in CAPITAL letters, ﬁrst name ﬁrst): _E_€ﬁ:j, _ _
NAME(sign): _ .
ID#: _ _
Instructions: There are four problems. Make sure that you have all 4 problems. You must show
all your work to receive full credit. Do not
Math 135A
January 11, 2016
B AYES
Theorem Suppose F1 , F2 , . . . , Fn are pairwise disjoint and
F1 F2 Fn = . Then, for any event A,
P(A) = P(A|F1 )P(F1 ) + P(A|F2 )P(F2 ) + + P(A|Fn )P(Fn ).
Math 135A
January 11, 2016
E XAMPLE
Ex Roll a die, and select a
Math 135A
January 11, 2016
J UST A C LASSIC P ROBLEM
Ex Youre on a game show where youre given the choice
between 3 doors, one of which contains a prize. You choose a
door; then the host opens another door and shows you that it is
empty. Youre now given a
Math 135A
January 6, 2016
P RINCIPLES OF C OUNTING
If rst you have n choices and then you have m choices,
then you have mn choices altogether.
The number of ways to choose k objects in order from a set
n!
of n is
.
(n k)!
The number of ways to choose k un
HW: ASSIGNMENT 1
Due: Tuesday, October 4th
Warm Up: You should also do the five Problems in Section 2 of the notes.
1 ) Ten people park their bikes outside the bar. Suppose everyone drinks
too much and takes a bike home at random. (1) How many ways can th
Math 135A, Winter 2014.
Homework 1, due Tue., Jan. 21
1. Fifteen married couples are at a dance lesson. Assume that (male-female) dance pairs are assigned
at random. (a) What is the number of possible assignments? (b) What is the probability that each
hus
Math 135A, Winter 2014.
Homework 7, due Tue., Mar. 17
1. Joint density of (X, Y ) is given by
f (x, y ) = xex(y+1) ,
x, y > 0.
(a) Find the conditional density of Y given X = x. (b) Compute the density of Z = XY .
2. Assume that X1 , . . . , X10 are indep
Math 135A, Fall 2013.
Homework 4, due Tue., Nov. 5
1. Roll a pair of fair dice successively, and each time observe the sum rolled. What is the probability
of getting two 7s before getting six even numbers?
2. A bag contains 8 white, 4 black and 2 grey bal
Math 135A, Fall 2013.
Homework 2, due Tue., Oct. 15
1. Three tours, A, B, and C, are oered to a group of 100 tourists. It turns out that 28 tourists sign
for A, 26 for B, 16 for C, 12 for both A and B, 4 for both A and C, 6 for both B and C, and 2 for all
Math 135A, Fall 2013.
Homework 3, due Tue., Oct. 22
1. There are three bags: A (contains 2 white and 4 red balls), B (8 white, 4 red) and C (1 white 3
red). You select one ball at random from each bag, observe that exactly two are white, but forget
which
Math 135A, Fall 2013.
Homework 5, due Tue., Nov. 12
1. Assume that weekly sales of diesel fuel at a gas station are X tons, where X is a random variable
with distribution function
c(1 x)4 0 < x < 1,
f (x) =
0
otherwise.
(a) Compute c. (b) Compute EX . (c)