Math 375 Review Lesson (Class of February 9th)
Correction of Problem 10
(a) 8! (basic permutation).
(b) 7.2.6!
Stage 1: How many arrangements for persons A and B? 7 (1st and 2nd seats; 2nd and 3rd seats etc.).
Stage 2: For each arrangement of persons A an

Math 375 Assignment #1: Solutions of the Sausage Problem
John has four children and four (indistinguishable) sausages. How many ways can he distribute the
sausages to the four children?
1. Answer the question by enumerating the possibilities (you dont hav

Math 375
Name:
The following problems have been excerpted from and/or inspired by
A First Course in Probability and Statistics by Peggy Tang Strait,
and Probability and Statistics by Ronald Rothenberg, unless otherwise noted.
Extra Credit : Fun with Covar

Math 375
Name:
The following problems have been excerpted from and/or inspired by
A First Course in Probability by Sheldon Ross, 8th Edition,
A First Course in Probability and Statistics by Peggy Tang Strait,
and Probability and Statistics by Ronald Rothe

Computations involving joint pmfs and pdfs
5. Suppose we have the joint density of X and Y given by f (x, y) =
(1/2)x2 y + (1/3)y, 0 < x < 2, 0 < y < 1 and f (x, y) = 0 elsewhere.
(10 points each)
(a) Determine the probability P (0 < X < 1, 0 < Y < 1).
(b

Math 375 Assignment
Try to Finish it for Sunday, February 14 (the solution will be posted on Blackboard
on Monday)
Force yourself to write and explain your reasoning clearly. As always, clarity of exposition will be a significant fraction of the grade (th

Math 375
Name:
The following problems have been excerpted from and/or inspired by Chapter 4 of
A First Course in Probability by Sheldon Ross, 8th Edition and
A First Course in Probability and Statistics by Peggy Tang Strait, 2nd Edition,
unless otherwise

Homework 4
Problem 1: Let be the collection of all real numbers. Fix any real number .
Define P as follows: given an event E,
1 if E
P (E) =
0 if 6 E
For example, P1 (0, ) = 1, while P1 (2, ) = 0, and for example, P (1, 1) = 0, while P (4, 4) = 1.
In this

Chapter 1: Combinatorial Analysis
Concept
Formula
Counting
Total number of outcomes
for r performed experiments,
each one resulting in ni outcomes.
Ordering
Number of different ways of ordering
r elements chosen among n elements.
Lottery
(How many sequenc

Homework 1
Problem 1: Prove that
n
n
n
n
+
+
+ . +
= 2n
0
1
2
n
Hint: Expand (1 + x)n by the binomial theorem then evaluate at some value of x.
Problem 2:
Prove that,
n+m
n m
n
m
n m
=
+
+ . +
k
k
0
k1
1
0
k
Hint: (1 + x)n+m = (1 + x)n (1 + x)m

1. Probability Measures
Let be any set, typically thought as the sample space of some random experiment.
Definition: A probability measure on , denote it as P , is a rule that assigns
a number P (E) for every event (subset of ) E and it satisfies the foll

Homework 6
Problem 1: Let X, Y be two independent simple random variables with mass functions,
t
0
1
3
fX 0.2 0.3 0.5
fY 0.1 0.8 0.1
Determine the mass function fX+Y and the probability P (XY = 0).
Problem 2: Let X Bin(n, p) and Y Bin(m, p), independent,

Homework 9
Problem 1: Let X Exponential(a), where a > 0, see HW8P12.
Show that P (X > t1 + t2 | X > t1 ) = P (X > t2 ) , here t1 and t2 are positive real numbers.
This property has an important consequence to physics. Let X be the time of decay of a radio

Answer 1: Consider the following table,
X Y X +Y
0 0
0
0 1
1
0 3
3
1 0
1
1 1
2
1 3
4
3 0
3
3 1
4
3 3
6
We see that the random variable X + Y takes on values: 0,1,2,3,4,6.
Now we compute, (the equation P (X = a, Y = b) = P (X = a)P (Y = b) follows from

Homework 4
Problem 1: Let be any (non-empty) set. Fix any sample point .
Define as follows: given an event E,
1 if E
(E) =
0 if 6 E
In this exercise show that is a probability measure.
This probability measure is referred to as the Dirac measure at .
Pro

Homework 7
Problem 1: Let X be a discrete random variable with the following mass function,
t
1
2
3
4
5 .
fX (t) 21 22 23 24 25 .
Compute: (i) the expected value E[X] (ii) the variance Var(X).
Problem 2: A drug has probability of .01 of failing to react w

Homework 5
Problem 1: Let X be a simple random variable on some sample space. Let f (t) = P (X = t) be the
mass function of X. Denote by F (t) = P (X t) the distribution function of X. Draw the graph of
F (t) given the following table for the mass functio

Answer 1: Since is in , then by definition () = 1.
If E, F are two events which do not contain , then EF does not contain . In particular, (EF ) = 0.
But we also have (E) = 0 and (F ) = 0. Therefore, (E F ) = (E) + (F ).
If E contains then F cannot contai

Answer 1:
Answer 2:
(i) Since X can only take on certain values write,
(0 X 2) = (X = 0) (X = 1) (X =
2)
Therefore,
P (0 X 2) = P (X = 0) + P (X = 1) + P (X = 2) = 0.2 + 0.3 + 0.4
Here we used the fact that (X = 0), (X = 1), and P (X = 2) are disjoint eve

Answer 1: Let us develop two series formulas.
Since
1
1x
=
P
n=0
xn for |x| < 1, it means, by differentiating,
X
1
=
nxn1
2
(1 x)
n=1
Now multiply both sides by x, to obtain, and then differentiate both sides again,
X
X
1+x
x
n
=
nx
=
=
n2 xn1
2
3
(1 x)
(

Math 375
Name:
The following problems have been excerpted from and/or inspired by
A First Course in Probability by Sheldon Ross, 8th Edition,
Introduction to Probability by Bertsekas and Tsitsiklis,
and Probability and Statistics by Ronald Rothenberg, unl

Homework 4
Problem 1: Let be the collection of all real numbers. Fix any real number .
Define P as follows: given an event E,
1 if E
P (E) =
0 if 6 E
For example, P1 (0, ) = 1, while P1 (2, ) = 0, and for example, P (1, 1) = 0, while P (4, 4) = 1.
In this

Joint Probability Functions
Functions of Random Variables, Expectation, and MGFs
Math 37500: Elements of Probability Theory
Volume 7 : Multiple Random Variables
Eli Amzallag
City College
1 / 88
Joint Probability Functions
Functions of Random Variables, Ex

Conditional Probability
Bayes Theorem and Conditioning
Independence
Math 37500: Elements of Probability Theory
Volume 3 : Conditional Probability and Independence
Eli Amzallag
City College
1 / 42
Conditional Probability
Bayes Theorem and Conditioning
Inde

Expectation and Variance Calculations
Poisson Processes Proofs
Math 37500: Elements of Probability Theory
Volume 5a :
Random Variables Addendum
Eli Amzallag
City College
1 / 31
Expectation and Variance Calculations
Poisson Processes Proofs
Outline
1
Expec

Continuous Random Variables
Expectation and Variance in the Continuous Case
Cumulative Distribution Functions of Continuous RVs
Several Important Continuous Distributions
Math 37500: Elements of Probability Theory
Volume 5 : Continuous Random Variables
El

The Normal Distribution
Distribution of a Function of a Random Variable
Conditional Densities
Chebyshevs Inequality and Moment Generating Functions
Math 37500: Elements of Probability Theory
Volume 6 :
The Normal Distribution, Conditional Distributions, a

What is Probability? : Four Points of View
Elements of Probability Theory
Probability Basics
Math 37500: Elements of Probability Theory
Volume 1 : Probability Basics
Eli Amzallag
City College
1 / 66
What is Probability? : Four Points of View
Elements of P