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Pairs of Random Variables

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.

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Pairs of Random Variables

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  1. Pairs of Random Variables Random Process

  2. 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

  3. Two Random Variables • The mapping is written as to each outcome is S

  4. Example 1

  5. Example 2

  6. Two Random Variables • The events evolving a pair of random variables (X, Y) can be represented by regions in the plane

  7. 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

  8. Two Random Variables • The probability of product-form events is • Some two-dimensional product-form events are shown below

  9. 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

  10. Pairs of Discrete Random Variables

  11. 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

  12. 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

  13. Example 3

  14. 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

  15. The Joint Cdf of X and Y

  16. Example 4

  17. 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

  18. The Joint Pdf of Two Continuous Random Variables

  19. 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

  20. 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

  21. The Joint Pdf of Two Continuous Random Variables

  22. Example 5

  23. Example 5

  24. Example 6

  25. 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

  26. 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

  27. 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

  28. Joint Moments and Expected Values • The expected value of is given by • Sum of random variable

  29. 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

  30. 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

  31. Conditional Probability Case 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

  32. 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

  33. Conditional Probability • Example:

  34. 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

  35. Conditional Probability • Example: binary communications system

  36. Conditional Probability Case 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

  37. 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

  38. Conditional Probability • Example

  39. Conditional Expectation • The conditional expectation of Y given X = x is given by • When X and Y are both discrete random variables

  40. Conditional Expectation • In particular we have where

  41. Pairs of Jointly Gaussian Random Variables • The random variables X and Y are said to be jointly Gaussian if their joint pdf has form

  42. 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

  43. Regular Class: NEXT WEEK: QUIZ 1 Material: Chapter 1 to 5, Garcia’s book Duration: max 1 hour

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