Final Study Guide
A. Probability spaces, outcomes, events (discrete case)
04/28/2011
1. What is the probability that a hand of 5 cards from a standard deck:
a) contains exactly two aces?
b) contains all cards of same suit?
c) contains no pairs?
2. An expe
Math 3355 Test 3 Spring 2007 Britt
Unsupported work is given very little credit. Pleas work the problems in order
on the fronts of your paper. You may leave factorials in your answer.
1. Suppose that X is a continuous random variable whose probability d
Instructor's Solutions Manual
THIRD EDITION
Fundamentals of P ROBABILITY
WITH
STOCHASTIC PROCESSES
Saeed Ghahramani
Instructor's Solutions Manual
Third Edition
Fundamentals of
With
ProbabilitY
Stochastic Processes
SAEED GHAHRAMANI
Western Ne
Quiz 01/20/2011
Name
1. A discrete probability space is a
or
set, equipped with a
.
2. How is a pmf used to dene a probability measure?
3. Dene the following words as used in probability modeling:
experiment
outcome
event
4. Consider the experiment ip
Quiz 01/27/2011
Name
1. How is a pmf used to dene a probability measure?
2. Dene the following words as used in probability modeling:
experiment
outcome
event
3. A jar contains 5 red, 3 green and 2 black beads. One is taken out at random and then
witho
Quiz 02/03/2011
Name
1. Suppose G and S are events with non-zero probability. By denition,
P (G|S ) =
1. Events A and B are said to be independent if:
3. A jar contains 5 red, 3 green and 2 black beads. One is taken out at random and then
without replacin
Quiz 02/10/2011
Name
1. Events A and B are said to be independent if:
2. A random variable is
3. Three dice are rolled. Let X be the total of the faces showing. Find P (X 5).
Extra credit: If the following is given to Mathematica as input, what will the
Quiz 02/17/2011
Name
1. [5 pts] Let X be a random variable on a discrete probability space with pmf f : [0, 1].
The expected value of X , denoted E (X ), is dened as follows:
E (X ) :=
.
2. [10 pts] Suppose the ves and the sixes on two dice are painted ov
Quiz 02/24/2011
Name
1. [5 pts] Let X be a discrete random variable with possible values in the set B and pmf
F : B [0, 1]. Then
.
E (X ) =
xB
2. [10 pts] Suppose X is a binomial random variable with parameters n and p.
X can be interpreted as:
the number
Quiz 03/17/2011
Name
Suppose X is a continuous random variable with pdf f .
1. [5 pts]
P (a X b) =
.
t
2. [5 pts] The function F (t) := f (x) dx is called the
a) antiderivative of X ;
b) cumulative distribution function of X ;
c) improper integral of X ;
Lecture 23.
Some topics from 10.1
Fact. Expectation is linear: E(aX ) = aE(X ) for any constant a and any random variable
X and E(X + Y ) = E(X ) + E(Y ) for any random variables X and Y . Thus, for any
constants ai and random variables Xi ,
E(a1 X1 + . .
Lecture 24.
See Chapter 10, section 2.
Denition. Suppose X and Y are random variables on the same probability space. The
covariance of X and Y is:
Cov(X, Y ) := E
X E(X ) Y E(Y )
.
Fact. Cov(X, Y ) = E(XY ) E(X )E(Y ).
Fact. Var(aX + bY ) = a2 Var(X ) + b
MATH 33551
FINAL EXAM
Spring 2009
You may leave answers in terms of binomial coecients, when applicable. * SHOW ALL WORK * (17) 1. A 2007 study showed that 17% of the citizens of Hawaii are smokers. Let X be the number of smokers in a randomly selected gr
Lecture 18. Adding Random Variables
Suppose X and Y are discrete random variables on cfw_0, 1, 2, 3, . . .. Suppose f and g are
the corresponding pmfs:
P ( X = x) = f ( x) ,
P (Y = y ) = g (y ).
We assume there is no inuence of one variable upon the other
Quiz 03/31/2011
Name
The Exponential Distribution
A large population declines through the deaths of individuals in such a manner that at the end of each
unit of time the population remaining is a xed fraction the population at the beginning of that unit.
Quiz 04/06/2011
Name
Suppose a discrete random variable X takes the values cfw_3, 4, 5 with non-zero probability. All other values
have zero probability. Assume P (X = 3) = a, P (X = 4) = b and P (X = 5) = c.
1. Write the generating function GX (t) of X .
Quiz 04/13/2011
Name
Here is the joint distribution function for a pair of discrete random variables:
X =1 X =2
Y =1
Y =2
.1
.2
X=3
.1
.3
.2
.1
1. What is P (X + Y 4)
2. Make tables of the values of the marginal pmfs pX (x) and pY (y ) .
3. Are X and Y in