6
A REVIEW OF SET THEORY
1 Basic Definitions
We define a set A as a collection of well-defined objects (called elements
or members of A) such that for any given object x one can assert without
dispute that either x A (i.e., x belongs to A) or x 6 A but no
2.7 Discrete Distributions
In this example,
There are 10 independent trials (10 independent losses).
Each trial is a Bernoulli trial (only two possible outcomes either a loss exceeds 10
or does not exceed 10).
The probability of success (the probabilit
2.4 Percentiles, Mode, Skewness, and Kurtosis
2.4
2.4.1
Percentiles, Mode, Skewness, and Kurtosis
Percentiles and Median
For a random variable, X, the 100pth percentile is denoted by p , where 0 p 1. If X is
discrete, then the 100pth percentile is the sma
2.7 Discrete Distributions
Therefore, the number of ways to choose the king of spades, exactly one other king, and
exactly two queens from a deck of 52 cards is
! "! "! "! "
1 3 4 44
= 792
1 1 2
1
This is divided by the number of ways to choose any five c
2.7 Discrete Distributions
(iv) Similarly, we write N 2 as N > 1 and proceed as in (ii).
V ar[N |N 2] = V ar[N |N > 1]
= V ar[N 1 + 1 | N > 1]
= V ar[N 1 | N > 1]
= V ar[N ]
1p
=
p2
1
1
= ! "62
1
6
= 30
2.7.6
Negative Binomial
Description
The negative bin
2.7 Discrete Distributions
2.7.7
Poisson
Description
The Poisson distribution is a discrete probability distribution used to model the occurrences
of an event during a fixed interval, where the occurrences in disjoint intervals are independent. The interv
STAT 2857-Assignment 3
Problem 1. Let X be a continuous non-negative random variable with density
function f, and let Y := X n . Find f Y , the probability density function of Y .
Problem 2. A system consisting of one original unit plus a spare can functi
STAT 2857 Assignment 4
Problem 1. a) Find the moments of the random variable X if its moment generating function is
MX (t) = (1 p1 p2 ) + p1 et + p2 e2t .
b) What is the variance of X?
Problem 2. Find the probability P (X 1.23) if X has moment generating
STAT 2857 Assignment 5
Problem 1. Let X have a geometric distribution with f (x) = p(1 p)x , x = 0, 1, 2, . . . Find the
probability function of R, the reminder when X is divided by 4.
Problem 2. The joint probability mass function of (X, Y ) is given by
2.1 Random Variables
SECTION 2: UNIVARIATE PROBABILITY DISTRIBUTIONS
In Section 1, general probability concepts were covered. We will use these concepts in this
section where we introduce random variables. Then, we will use random variables as a basis
for
1.1 Basic Probability Concepts
SECTION 1: GENERAL PROBABILITY
Have you read the Introduction?
The Introduction will help you get started correctly by guiding you with:
Mission statement of the book
Study schedule and tips
Other helpful resources includ
1.3 Bayes Theorem
1.3
Bayes Theorem
Bayes Theorem is also known as Bayes rule. It is derived from the Law of Total Probability
and conditional probability.
The Law of Total Probability from Section 1.1.2 states that if A1 , A2 , ., An is a partition of
th
2.3 Moment Generating Functions
2.3 Moment Generating Functions
A moment generating function (MGF) is a function that generates the moments of the random variable associated with the MGF. For example, the moment generating function of the
random variable
1.2 Conditional Probability and Independence
1.2
1.2.1
Conditional Probability and Independence
Conditional Probability
Conditional probability is the probability that an event occurs based on a condition, or given
that another event has occurred. In othe
2.5 Chebyshevs Inequality
2.5 Chebyshevs Inequality
The Chebyshevs inequality is also spelled as Tchebyches inequality. It is an inequality
1
which states that for any probability distribution, at most 2 of the values in the distribuk
tion can be more tha
1.4 Counting Techniques and Combinatorial Probability
1.4
Counting Techniques and Combinatorial Probability
This section introduces the branch of mathematics known as combinatorics. For the purpose
of probability theory within the scope of the exam syllab
2.7 Discrete Distributions
2.7 Discrete Distributions
There are a number of discrete distributions that are commonly used in Exam P problems,
some more frequently than others. In this section, we will cover the following seven discrete
distributions:
1.
2
2.7 Discrete Distributions
2.7.3
Binomial
Description
A binomial distribution is a discrete probability distribution of the number of successes in
a sequence of n independent Bernoulli trials, each of which yields success with probability
p.
If X is a bin
2.6 Univariate Transformations
2.6
Univariate Transformations
A transformation of a random variable uses the distribution function of a random variable
to determine the distribution function of another random variable that is related to the first
via an e
2.7 Discrete Distributions
2.7.2
Bernoulli
Description
The Bernoulli distribution is a discrete probability distribution that only has two values, 0
and 1. The probability of 1, which is sometimes called success, is p. The probability of 0,
which is somet
2.2 Moments
2.2
2.2.1
Moments
Expected Value
Density Function Method
If X is a discrete random variable, then the mean, or the expectation, or the expected value,
or first moment of X, denoted as E[X], is
E[X] =
!
all x
x Pr(X = x)
If X is a continuous ra