pairs of random variables random process. introduction in this lecture you will study: joint pmf,...
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Pairs of Random Variables
Random Process
Introduction In this lecture you will study:
Joint pmf, cdf, and pdf Joint moments The degree of “correlation” between two random
variables Conditional probabilities of a pair of random
variables
Two Random Variables The mapping is written as to
each outcome is S
Example 1
Example 2
Two Random Variables The events evolving a pair of random
variables (X, Y) can be represented by regions in the plane
Two Random Variables To determine the probability that the pair
is in some region B in the plane, we have
Thus, the probability is
The joint pmf, cdf, and pdf provide approaches to specifying the probability law that governs the behavior of the pair (X, Y)
Firstly, we have to determine what we call product form
where Ak is one-dimensional event
Two Random Variables The probability of product-form events is
Some two-dimensional product-form events are shown below
Pairs of Discrete Random Variables Let the vector random variable
assume values from some countable set The joint pmf of X specifies the probabilities of
event
The values of the pmf on the set SX,Y provide
Pairs of Discrete Random Variables
Pairs of Discrete Random Variables The probability of any event B is the sum of
the pmf over the outcomes in B
When the event B is the entire sample space SX,Y, we have
Marginal Probability Mass Function The joint pmf provides the information about
the joint behavior of X and Y The marginal probability mass function shows
the random variables in isolation
similarly
Example 3
The Joint Cdf of X and Y The joint cumulative distribution function of X
and Y is defined as the probability of the event
The properties are
The Joint Cdf of X and Y
Example 4
The Joint Pdf of Two Continuous Random Variables Generally, the probability of events in any
shape can be approximated by rctangles of infinitesimal width that leads to integral operation
Random variables X and Y are jointly continuous if the probability of events involving (X, Y) can be expressed as an integral of probability density function
The joint probability density function is given by
The Joint Pdf of Two Continuous Random Variables
The Joint Pdf of Two Continuous Random Variables The joint cdf can be obtained by using this
equation
It follows
The probability of rectangular region is obtained by letting
The Joint Pdf of Two Continuous Random Variables We can, then, prove that the probability of an
infinitesimal rectangle is
The marginal pdf’s can be obtained by
The Joint Pdf of Two Continuous Random Variables
Example 5
Example 5
Example 6
Independence of Two Random Variables X and Y are independent random variable if
any event A1 defined in terms of X is independent of any event A2 defined in terms of Y
If X and Y are independent discrete random variables, then the joint pmf is equal to the product of the marginal pmf’s
Independence of Two Random Variables If the joint pmf of X and Y equals the product of
the marginal pmf’s, then X and Y are independent
Discrete random variables X and Y are independent iff the joint pmf is equal to the product of the marginal pmf’s for all xj, yk
Independence of Two Random Variables In general, the random variables X and Y are
independent iff their joint cdf is equal to the product of its marginal cdf’s
In continuous case, X and Y are independent iff their joint pdf’s is equal to the product of the marginal pdf’s
Joint Moments and Expected Values The expected value of is given by
Sum of random variable
Joint Moments and Expected Values In general, the expected value of a sum of n
random variables is equal to the sum of the expected values
Suppose that , we can get
Joint Moments and Expected Values The jk-th joint moment of X and Y is given by
When j = 1 and k = 1, we can say that as correlation of X and Y
And when E[XY] = 0, then we say that X and Y are orthogonal
Conditional ProbabilityCase 1: X is a Discrete Random Variable For X and Y discrete random variables, the
conditional pmf of Y given X = x is given by
The probability of an event A given X = xk is found by using
If X and Y are independent, we have
Conditional Probability The joint pmf can be expressed as the product
of a conditional pmf and marginal pmf
The probability that Y is in A can be given by
Conditional Probability Example:
Conditional Probability Suppose Y is a continuous random variable,
the conditional cdf of Y given X = xk is
We, therefore, can get the conditional pdf of Y given X = xk
If X and Y are independent, then The probability of event A given X = xk is
obtained by
Conditional Probability Example: binary communications system
Conditional ProbabilityCase 2: X is a continuous random variable If X is a continuous random variable then P[X
= x] = 0 If X and Y have a joint pdf that is continuous
and nonzero over some region of the plane, we have conditional cdf of Y given X = x
Conditional Probability The conditional pdf of Y given X = x is
The probability of event A given X = x is obtained by
If X and Y are independent, then and
The probability Y in A is
Conditional Probability Example
Conditional Expectation The conditional expectation of Y given X = x is
given by
When X and Y are both discrete random variables
Conditional Expectation In particular we have
where
Pairs of Jointly Gaussian Random Variables The random variables X and Y are said to be
jointly Gaussian if their joint pdf has form
Lab assignment In group of 2 (for international class: do it personally),
refer to Garcia’s book, example 5.49, page 285 Run the program in MATLAB and analyze the result Your analysis should contain:
The purpose of the program Line by line explanation of the program (do not copy from
the book, remember NO PLAGIARISM is allowed) The explanation of Fig. 5.28 and 5.29 The relationship between the purpose of the program and
the content of chaper 5 (i.e. It answers the question: why do we study Gaussian distribution in this chapter?)
Try using different parameter’s values, such as 100 observation, 10000 observation, etc and analyze it
Due date: next week
Regular Class:
NEXT WEEK: QUIZ 1
Material: Chapter 1 to 5, Garcia’s book
Duration: max 1 hour
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