Table of Contents
Assigned Reading
Although our text does not conduct a formal probability review, it would be advised that you consult your prerequisite course's probability textbook. The free, online Khan Academy also has a nice collection of short probability and statistics tutorials covering a wide variety to topics applicable to this course.
Useful Equations
Rules of Integration
Chain Rule for Integration
Rules of Differentiation
Chain Rule for Differentiation
Laws for Probabilities of Events
Venn Diagram
Law of Addition
Definition of Conditional Probability
Bayes' Rule
Random Variables (RVs)
Conditional Probability
If X and Y are discrete RVs, the definition of conditional probability for all values of y such that Pr(Y = y) > 0 is
If X and Y are continuous RVs and have joint PDF fXY(x,y), then the conditional PDF for all values of y such that fY(y) > 0 is
For RV X, we can find Pr(X <= x) by conditioning on RV Y. If Y is a discrete RV, then
If Y is a continuous RV with desity fY(y), then
Conditional Expectation
If X and Y are jointly discrete RVs, then for all values of y, such that Pr(Y = y) > 0,
Likewise, if X and Y are jointly continuous RVs, then for all values of y such that fY(y) > 0,
Computing Expectations by Conditioning
For RV X, we can find E[X] by conditioning on RV Y. If Y is a discrete RV, then
If Y is a continuous RV with density fY(y), then
Integer Series Identities
If 0 < p < 1, then
If a != 1, then
If a != 1, then
where, k factorial = k! = (k)(k - 1)(k - 2)(...)(1).
Combinations of Choosing k from n Elements
Frequently Used Random Variables
Exponential Continuous Random Variable
RV X with parameter α > 0 that supports [0, ∞)
Memoryless property:
Uniform Continuous Random Variable
RV X that supports [a, b]
Bernoulli Discrete Random Variable
RV X with parameter 0 < p < 1 that supports k = {0, 1}
Binomial Discrete Random Variable
RV X with parameters positive integer n and 0 < p < 1 that supports k = {0, 1, ..., n}
Geometric Discrete Random Variable
RV X with parameter 0 < p < 1 indicating the probability of success.
The number X of Bernoilli trials needed to get one success, supported on the set k = {1, 2, ...}
Memoryless property:
The number of Y = X - 1 failures before the first success, supported on the set k = {0, 1, ...}
Poisson Discrete Random Variable
RV X with parameter λ > 0 that supports k = {0, 1, ...}
Review of Random Variables (RVs)
Cumulative Distribution Function
A RV X is defined by a function FX(x), called a cumulative distribution function (CDF) of X, which has the following properties:
Example
CDF of a RV:
Probability
For a RV X and any value a, FX(a) represents the probability that X is less than or equal to a, i.e.:
Probability Space
FX(x) defines a probability space, and
Continuous RVs
If the CDF of FX(x) of the RV X is a continuous function of x, then X is a continuous RV.
Example
CDF of a continuous RV:
CDF of a non-continous RV:
Probability Density Function
If a RV X is continuous, and there is a function fX(x), called the probability density function (PDF), such that:
then X is an absolutely continuous RV.
If X is differentiable at x, then:
The following are properties of a PDF:
Example
PDF of a continuous RV that is uniformly distributed on the interval [0,C]:
PDF of a continuous RV that is exponentially distributed with parameter α > 0:
