Tum Mt'Gmw-H 1W
Probability,
Statistics and
r Random Processes
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CE/EE 302:
Probabilistic Methods in Electrical and Computer Engineering
Chapter 4 : One Random Variable
Week 8 Week 9 Lectures: 1-2
Dr. Pierre E. ABI-CHAR
Lecture Outline
I. The Cumulative Distribution Function
II. The Probability Density Function
III. Th
Basic Concepts.
SetAis asubsetof setBif every element ofAis also an
element
ofB.
We writeAB, where " indicates "subset." Equivalently,
we sayBis asupersetofA, orBA.
TWO Important Laws
EXAMPLE
SOLUTION
Cartesian product
Cardinality: Countable and
Uncountab
Lectures (10-11)
Probability and statistics
MTHN203
Jointly distributed Random variables
Dr. Mohamed Ibrahim Wafa
2
Jointly distributed Random
variables
Multivariate distributions
2 or more random variables (X, Y, etc)
defined for the same random experime
1
Jointly distributed Random
variables
Multivariate distributions
2
Jointly distributed Random
variables
Multivariate distributions
2 or more random variables (X, Y, etc)
defined for the same random
experiment.
3
(introduction)Example for illustration
To
1
Previous lecture
2
Sample space
This is denoted with an S and is a set
whose elements are all the possibilities
that can occur
Each element
of S is called
an outcome.
A probability of an outcome is a number and
has two properties:
1. The probability ass
1
Introduction to
probability
Probability
Introduction
Probability theory is a branch of mathematics
for dealing with uncertainty
2
Why Probability?
Plays an important role in decision making in
day-to-day activities (Weather, .).
How do the insurance com
1
Previous lecture
2
Rule of conditional probability
P( A B)
P( A | B)
P( B)
Similarly
P (A B )
P (B | A )
P (A )
3
Note
P( A B)
P( A | B)
P( B)
P (A B ) P (A | B )P (B )
Similarly
P ( A B ) P ( B | A) P ( A)
Therfore
P ( A | B) P( B) P( B | A) P( A)
4
1
Normal Probability Distributions
x
The normal distribution is a descriptive model
that describes real world situations.
The mean
Bell shaped and is symmetric about the mean
The total area that lies under the curve is one
2
3
The Normal Distribution:
continuous probability
distributions
Uniform distribution
Exponential distribution
Normal distribution
Uniform Distribution
A Uniform Distribution has equally likely
values over the range of possible outcomes.
A graph of the uniform probability distributi
Lecture (9)
Probability and statistics
MTHN203
Continuous Probability
Distributions
Dr. Mohamed Ibrahim Wafa
1
Continuous Probability
Distributions
Uniform distribution
Exponential distribution
Normal distribution
Uniform distribution(introduction)
The fi
Basic Statistics
1. Introduction
One way to characterize statistics is
that:
Statistics is science for developing
methods to derive effectively
information from numerical data.
In short Statistics is a science of
information.
Statistical analysis usually
Central Limit Theorem
Sampling Distributions
To make inferences about a population, we need to
understand sampling
The sample mean varies from sample to sample
The sample mean has a distribution; we need to
understand how the sample mean varies and the
1
Normal Probability Distributions
x
The normal distribution is a descriptive model
that describes real world situations.
The mean
Bell shaped and is symmetric about the mean
The total area that lies under the curve is one
2
3
The Normal Distribution:
1
Sets(Revision)
2
Sets(Revision)
A set is a collection of objects.
The objects in a set are called elements of the set.
Examples:
Natural numbers N = cfw_0, 1, 2, 3,
Integers Z = cfw_, -2, -1, 0, 1, 2,
Positive Integers Z+ = cfw_1, 2, 3, 4,
Rea
1
Conditional Mean and Variance
The conditional mean of Y given X = x,
E (Y X ) Y X yfY X ( y )
y
And the conditional variance of Y given X = x
V (Y X ) 2Y X ( y Y X ) 2 fY X ( y )
y
V (Y X ) 2Y X y 2 fY X ( y ) 2Y X
y
Similarly
E ( X Y ) X Y xf X Y ( x)
Topic #3
Random Variables &
Discrete Distributions
Contd
1
Discrete Distributions
Binomial Distribution
Examples
1) Flip a coin 10 times.
Number of trials
n = 10
Each trial has only two possible Head = success
outcomes
Tail = failure
Prob. of success & fa
Topic #1
Introduction
1
What is Statistics?
Remember?
Statistics is a way to get information from data
Statistics
Data
Information
Statistics is the science that deals with collecting,
summarizing, analyzing, interpreting, and
presenting data, and drawing
Name: Dr. Nesma Saleh
E-mail: [email protected]
Office Hours: Sunday: 11:30 1:00
Tuesday: 11:30 1:00
By Appointment via E-mail
Office: 85 4th Floor Old Building
Recall in S101 course when you were taught that
Statistics has two main branches: (1) Desc
Topic #3
Random Variables &
Discrete Distributions
1
Random Variables
What is a random variable?
A random variable is a way to quantify (or assign
numbers) all the outcomes in the sample space of a
random experiment.
A random variable is usually denoted
Topic #3
Random Variables &
Discrete Distributions
Contd
1
Probability Distribution
A probability distribution is a table, graph, or a
function that gives all possible values of the random
variable and the probability associated with each
value.
Example
C
Topic #3
Random Variables &
Discrete Distributions
Contd
1
Discrete Distributions
Properties of the Discrete Distribution
Calculating the variance and standard deviation
Recall: variances depend on the squared deviations
of the observations from its mean
Topic #5
Sampling
Distribution
1
(1) Sampling Distribution of
The distribution of the sampling mean
Since now follows the Normal distribution with
parameters and , then accordingly
Z
X X
X
~ N (0 ,1)
Sampling from an infinite population or finite with
re
Topic #5
Sampling
Distribution
1
(1) Sampling Distribution of
In our study to the sampling distribution of , we will consider
the following sampling scenarios:
Sampling may be from
Infinite Population
No difference between
replacement or
without replacem
Topic #2
Probability
Contd
1
Counting the Possibilities
3. Sampling from a Population without Replacement
If we select n items from a population of size N
without replacement, then we use:
Permutations if the order is important.
Combinations if the ord
Topic #2
Probability
1
What is Probability?
Probabilities play a vital role in our daily life.
Probabilities are used in measuring uncertainty.
We usually face situations when we need to take a
decision however based on incomplete information.
In such
Topic #2
Probability
Contd
1
How to Assign (Calculate) Probability?
Method (1): Classical Approach
Properties of the Classical Method
The probability can be determined
performing the experiment.
before
We know the possible outcomes of the experiment
with