Lecture Notes for Introductory Probability
Janko Gravner
Mathematics Department
University of California
Davis, CA 95616
[email protected]
January 5, 2014
These notes were started in January 2009 with help from Christopher Ng, a student in
Math 135
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
First Course in Probability, A
2/9/17, 2:44 PM
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prior permission. Violators will be prosecuted.
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First Course in Probability, A
3/8/17, 10:51 AM
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prior permission. Violators will be prosecuted.
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Math 135A January 25, 2016
POISSON APPROXIMATION TO BINOMIAL
El 5 _= If ORG Math 135A January 25, 2016
WHY?
U(\S Awkc/I <02:be " W J Rx?
Cm - (CLCU KW a? rK a.) 245.
/
Wog
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
First Course in Probability, A
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prior permission. Violators will be prosecuted.
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First Course in Probability, A
3/1/17, 10:54 PM
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prior permission. Violators will be prosecuted.
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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
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
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
Math 135A, Fall 2013.
Homework 6, due Tue., Nov. 19
1. A fair die is rolled ve times. (a) Compute the joint p.m.f. of X and Y , where X is the rst number
rolled and Y is the largest of the ve numbers rolled. (b) Compute the joint p.m.f. of X and Y , where
Math 135A, Fall 2013.
Homework 7, due Tue., Dec. 3
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 independ
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