Homework # 6 Solutions
Enrique Campos-Nnez a
The George Washington University
ApSc 116
1/9
Problem (Page 175 # 1, Section 3.9)
Suppose that X1 and X2 are i.i.d. random variables and that each of them has a uniform distribution on the interval [0, 1]. Find

Homework # 3 Solutions
The George Washington University
ApSc 116
1 / 10
Problem (Page 102, # 3)
Suppose that two balanced dice are rolled, and let X denote the absolute value of the dierence between the two numbers that appear. Determine and sketch the p.

Homework # 4 Solutions
The George Washington University
ApSc 116
1 / 11
Problem (Page 136, # 7)
Suppose that the joint p.d.f. of X and Y is as follows: f (x , y ) = 2xe y 0 for 0 x 1 and 0 < y < , otherwise.
Are X and Y independent? Solution: Since f (x ,

Introduction to Probability
Enrique Campos-Nnez a
The George Washington University
ApSc 116
Table of Contents
Interpretations of Probability
Probability and Set Theory
The Denition of Probability
Counting Methods
Interpretations of Probability
Interpretat

Conditional Probability
Enrique Campos-Nnez a
The George Washington University
ApSc 116
Table of Contents
Conditional Probability
Event Independence
Independence and Conditional Probabilities
Bayes Theorem
The Denition of Conditional Probability
The quant

Random Variables and Probability Distributions
Enrique Campos-Nnez a
The George Washington University
ApSc 116
Table of Contents
Random Variables
Common Discrete Distributions
Continuous Random Variables and Distributions
The Distribution Function
Random

Multivariate Distributions
Enrique Campos-Nnez a
The George Washington University
ApSc 116
Multivariate Data: Height, Weight, and Fat
Motivating multivariate distributions
Table of Contents
Bivariate Distributions
Conditional Distributions
Multivariate Di

Functions of Random Variables
Enrique Campos-Nnez a
The George Washington University
ApSc 116
1 / 32
Table of Contents
Functions of a Single Random Variable
Functions of Several Random Variables
Sum of Random Variables
2 / 32
Functions of a Single Random

Expectations
Enrique Campos-Nnez a
The George Washington University
ApSc 116
1 / 41
Table of Contents
The Expectation of a Random Variable Expectations and Joint Distributions Properties of the Expectations Expectation of Non-negative r.v.s Variance Momen

Discrete Distributions
Enrique Campos-Nnez a
The George Washington University
ApSc 116
Campos-Nnez (GWU) a
Discrete Distributions
ApSc 116
1 / 32
Table of Contents
1
Bernoulli Binomial Hypergeometric Distribution Poisson Distribution Negative Binomial The

Homework # 5 Solutions
The George Washington University
ApSc 116
1 / 11
Problem (Page 157 # 1, Section 3.7)
Suppose that three random variables X1 , X2 , and X3 have a continuous joint distribution with the following joint p.d.f.: ( c (x1 + 2x2 + 3x3 ) 0

Homework # 1 Solutions
ApSc 116
Problem Page 18 # 7
We know Pr (AB ) = Pr (A) + Pr (B ) Pr (A B ) = 0.4 + 0.7 Pr (A B ) = 1.1 Pr (A B ) Since 0.7 Pr (A B ) 1 (the latter is true if A B ), we can conclude 0.1 Pr (AB ) 0.4.
Problem Page 27 # 8
There are 75

Homework # 2 Solutions
The George Washington University
ApSc 116
1/8
Problem (Page 55 # 9)
Suppose that a box contains one blue card and four red cards, which are labeled A, B, C, and D. Suppose also that two of these ve cards are selected at random, with

Functions of Random Variables
Lecture 4
Spring 2002
Function of a Random Variable
Let U be an random variable and V = g (U ). Then V is also a rv since, for any outcome e, V (e) = g (U (e). There are many applications in which we know FU (u) and we wish t

Midterm Exam Solutions Ap.Sc. 116
Problem 1. Suppose that a box contains ve coins, and that for each coin there is a dierent probability that a head will be obtained when the coin is tossed. Let pi denote the probability of a head when the ith coin is tos

Transformations and Expectations of random variables
X FX (x): a random variable X distributed with CDF FX . Any function Y = g (X ) is also a random variable. If both X , and Y are continuous random variables, can we nd a simple way to characterize FY an

1
Lecture 1. Transformation of Random Variables
Suppose we are given a random variable X with density fX (x). We apply a function g to produce a random variable Y = g (X ). We can think of X as the input to a black box, and Y the output. We wish to nd the

Transformations of Continuous Random Variables I. Univariate case. If \ is a continuous random variable with pdf 0 and 1 is a continuous, invertible function, then the pdf for ] 1\ is given by 0] C l .B l 0 1" C .C Example 1. Let \ have pdf 0 given by 0 B

Special Discrete Random Variables Examples
How can I tell which variable to use? Define the variable and its range What are the variables smallest and largest values? Note that Binomial x= 0, , n Neg Binomial x = 0, 1, . (no upper limit) Poisson x = 0, 1