Probability
Basic Principle of Counting - if 1 experiment can result in m
outcomes & another can result in n, there are mn total outcomes
Permutations - arrangement of objects. There are n! ways of
arranging n objects. Ex permutation with grouping. 4 math
NOTES ON RANDOM VARIABLES AND THEIR DISTRIBUTIONS by A. Ledoan In the first three chapters of his book [1] Ross sets forth the formulation of probability theory. The idea of an experiment, a sample space corresponding to the experiment, and events in
Introduction to Probability Lecture Notes
Version July 28, 2016
David F. Anderson
Timo Seppalainen
Benedek Valko
c Copyright 2016 David F. Anderson, Timo Seppalainen and Benedek Valko
Contents
Preface
1
Chapter 1.
Random outcomes and random variables
5
1.
Math 201 - Homework 5
Exercise 1.
(a) Show that if X Geom(p) then
P (X = n + k|X > n) = P (X = k), for every n, k 1.
This one of the ways to define the memoryless property of the geometric distribution. It
states the following: given that there are no suc
Math 201 - Homework 3
Exercise 1.
An urn contains b red and c green balls. One ball is drawn randomly from
the urn and its color observed; it is then returned in the urn, and additional r balls of the
same color are added to the urn, and the selection pro
Math 201 - Homework 11
Exercise 1.
Let X have the density function
(
2x, if 0 < x < 1,
fX (x) =
0,
otherwise.
and let Y be uniformly distributed on the interval (1, 2). Assume X and Y are independent.
Give the joint density function of (X, Y ). Calculate
Math 201 - Homework 10
Exercise 1.
Let X1 , X2 be independent Poisson random variables such that Xi has mean
i for i = 1, 2. Define
Y = X1 + X2 .
Show that Y is distributed Poisson with mean 1 + 2 .
You can use the following fact: if W and Z are independe
Math 201 - Homework 2
Exercise 1. May and Jonny are playing paintball in opposite teams and they try to shoot
each other simultaneously.
The probability that both May and Jonny hit the target in a single shot is 1/8.
The probability that May hits the ta
Math 201 - Homework 7
Exercise 1.
Choose a point uniformly at random from the triangle with
vertices (0, 0), (0, 30), and (20, 30). Let (X, Y ) be the coordinates of the
chosen point.
(a) Find the cumulative distribution function of X.
(b) Use part (a) to
Math 201 - Homework 8
Exercise 1.
The homework submissions to the university computer center start at midnight (00:00). The number of homework submissions between midnight and any time t > 0
afterwords is distributed Poisson with mean t, where > 0 is some
MATH 201 - INTRODUCTION TO PROBABILITY
Homework Assignment 1
(1) A fair coin is tossed 11 times. In the end of the experiment we get an ordered vector of
length 11 with elements H and T.
(a) What is the number of possible outcomes?
(b) What is the number
Math 201 - Homework 12
Exercise 1. Customers arrive to the store according a Poisson process with intensity = 1
customers per hour. Let N [0, t] be the number of arrivals between [0, t].
(a) Find the probability
P (N [0, 2] = 2, N [2, 3] = 1|N [0, 6] = 6)
Math 201 - Homework 4
Exercise 1.
Suppose that a persons birthday is a uniformly random choice from the
365 days of a year (leap years are ignored), and one persons birthday is independent of the
birthdays of other people. Alex, Betty and Conlin are compa
Chapter 1
Combinations with replacement: n for 71 possibilities and 7 trials.
Permutations: n! for 71 unique objects of lm for k repetitions.
Let there be a committee of 3 men and 2 women out of 7 men and 5
women. 2 men wont sit together. Number of combin
HOMEWORK #1 MTH 201, FALL 2007 DUE: WEDNESDAY, SEPTEMBER 19, IN-CLASS AT THE BEGINNING OF CLASS The following problems are numbered the same in both the 6th and 7th edition of Ross's textbook. Chapter 1, Problems: 13, 22; Chapter 1, Theoretical Exe
NOTES ON CONDITIONAL PROBABILITY, BAYES' THEOREM, AND INDEPENDENCE by A. Ledoan The ideas of conditional probability and independence play a central role in the study of random phenomena. Conditional probability is the study of how additional informa