Chapter 8 Probability
Distributions
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8.1 Random variables
8.2 Probability distributions
8.3 Binomial distribution
8.4 Hypergeometric distribution
8.5 Poisson distribution
8.7 The mean of a probability distribution
8.8 Standard de

CIS 2033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Michael Baron. Probability and Statistics for Computer Scientists, CRC 2006
Instructor Longin Jan Latecki
Ch. 6 Simulations
What is a simulation?
One uses a model t

Section 3.4 Families of Discrete Distributions
All models are wrong, but some are useful.
George Box, statistician
Some probability distributions occur frequently enough that we can study the
distribution itself and later use it to model a situation.
Here

Nave Bayes Classifier
Ke Chen
http:/intranet.cs.man.ac.uk/mlo/comp20411
/
Modified and extended by Longin Jan Latecki
[email protected]
Probability Basics
Prior, conditional and joint probability
P(X )
Prior probability:
Conditional probability:
P(X1| X

CIS 2033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Instructor Longin Jan Latecki
Chapter 5: Continuous Random Variables
Probability Density Function of X
A random variable (RV) X is continuous if there exists functi

Section 2.2 Axioms and rules of probability
Here and elsewhere we shall not obtain the best insight into things until we actually see
them growing from the beginning
Aristotle, from the work Politics
Axioms of probability
A mathematically rigorous study o

Lecture 11: The Bernoulli and Binomial Distributions
1. Definitions
Definition: A random variable X is said to be a Bernoulli random variable with parameter p if it takes values in the set cfw_0, 1 with probability mass function
p(0) = P(X = 0) = 1 p
p(1)

Introduction to Probability
Example Sheet 1 - Michaelmas 2006
Michael Tehranchi
Problem 1. Show that if F is a sigma-field on a set then that both and are elements
of F.
Solution 1. If A is in F, so is the complement Ac . Hence the union A Ac = is in F, a

Binomial Distribution
And general
discrete probability distributions.
Random Variable
A random variable assigns a number to a
chance outcome or chance event.
The definition of the random variable is
denoted by uppercase letters at the end of
alphabet, s

CIS 2033 based on
Dekking et al. A Modern Introduction to Probability and Statistics, 2007
Instructor Longin Jan Latecki
Chapter 7: Expectation and variance
The expectation of a discrete random variable X taking the values a1,
a2, . . . and with probabili

Sections 3.1 - 3.3 Random Variables
Go down deep enough into anything and you will find mathematics.
Dean Schlicter
The Definition of a Random Variable
A random variable is a function of an outcome:
Domain:
Range may be:
Finite
Infinite and countable
Unco

C4: DISCRETE RANDOM VARIABLES
CIS 2033 based on
Dekking et al. A Modern Introduction to Probability
and Statistics. 2007
Longin Jan Latecki
Discrete Random Variables
Discrete random variables (RVs) are obtained by counting and have sample spaces
which are

Sections 3.1 - 3.3 Random Variables
Go down deep enough into anything and you will find mathematics.
Dean Schlicter
The Definition of a Random Variable
A random variable is a function of an outcome:
Domain:
Range may be:
Finite
Infinite and countable
Unco

Section 3.4 Families of Discrete Distributions
All models are wrong, but some are useful.
George Box, statistician
Some probability distributions occur frequently enough that we can study the
distribution itself and later use it to model a situation.
Here

CIS 2033
Based on text book:
F.M. Dekking, C. Kraaikamp, H.P.Lopulaa, L.E.Meester. A
Modern Introduction to Probability and Statistics
Understanding Why and How
Instructor: Dr. Longin Jan Latecki
1
Chapter 15 Exploratory data analysis:
graphical summaries

Section 2.4 Conditional Probability
Section 2.4: Conditional Probability
If people do not believe that mathematics is simple, it is only because they do not
realize how complicated life is.
John Louis von Neumann
The conditional probability of event give

Ch. 17 Basic Statistical Models
CIS 2033: Computational Probability and Statistics
Prof. Longin Jan Latecki
Prepared by: Nouf Albarakat
Basic Statstcal Models
Random samples
Statstcal models
Distributon features and sample statstcs
Estmatng features of

UNR, MATH/STAT 352, Spring 2007
Binomial(n,p)
n k
n!
n k
P( X k ) p (1 p )
p k (1 p ) n k
( n k )! k !
k
E ( X ) np
np(1 p )
2
X
UNR, MATH/STAT 352, Spring 2007
0.1
0.2
0.3
Number of successes within 1
symmetric Bernoulli trial
can only be 0 or 1. Thes

Review of the
Binomial Distribution
By Young Jun Choi
Definition
The Binomial Distribution is the
distribution of _?_
Definition
The Binomial Distribution is the
distribution of COUNTS.
It counts the number of successes in a
certain number of trials.
F

Chapter 16
Random Variables
Ranya Kaluarachchi
George Weng
Period 3
Random Variables
A random variable assumes any of several different values as a result of some
random event.
Two Types:
Discrete random variables can take one of a FINITE number of DISTIN

Binomial
Distribution and
Applications
Binomial Probability
Distribution
A binomial random variable X is defined to the
number of successes in n independent trials
where the P(success) = p is constant.
Notation: X ~ BIN(n,p)
In the definition above notice

Ch. 21 Maximum Likelihood
CIS 2033: Computational Probability and
Statistics
Prof. Longin Jan Latecki
1
2
3
4
5
Since we model the data as a random sample from a geometric distribution, the
probability of the data for smokersas a function of pis given by

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1/7/10
9:45 AM
Page 28
IES 302: Engineering Statistics
HAPTER 2 PROBABILITY
2011/2
HW Solution 1 Due: February 1
Lecturer: Prapun Suksompong, Ph.D.
2-18. In a magnetic storage device, three attempts are made
ES FOR SECTION 2-1
to read data before an

Section 2.2 Axioms and rules of probability
Here and elsewhere we shall not obtain the best insight into things until we actually see
them growing from the beginning
Aristotle, from the work Politics
Axioms of probability
A mathematically rigorous study o

Section 4.2(a) Families of Continuous Distributions
We balance probabilities and choose the most likely. It is the scientific use of the
imagination.
Sherlock Holmes, The Hound of the Baskervilles
Here we review some commonly used continuous distributions

CIS 2033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Slides by Michael Maurizi
Instructor Longin Jan Latecki
C9: Joint Distributions and Independence
9.1 Joint Distributions of Discrete
Random Variables
Joint Distribu

Complement
If A is an event, then the complement of A, written Ac, means
all the possible outcomes that are not in A.
For example, if A is the event UNC wins at least 5 football
games, then Ac is the event UNC wins less than 5 football
games.
1
We can rep

Sections 3.1 - 3.3 Random Variables
Expected Value of a Random Variable
This is the same formula as in physics for the center of mass (or of gravity) of a system
of point masses ( ) located at the points .
Example: If a coin is tossed 3 times, what is the

Section 2.1 Events as sets
The whole of science is nothing more than a refinement of everyday thinking.
Albert Einstein, from the essay Physics and Reality
Three interpretations of probability (and statistics)
I.
We have an intuitive idea of probability a

CS/SE 3341
Last Name:
First Name:
The Final Examination
Show your work. No points will be given if the appropriate work is not shown, even if the
answer is correct.
1. A rat moves through the compartments of the maze shown in the following figure by choo