1. Multiple Random Variables Two Discrete Random Variables –Joint pmf –Marginal pmf Two Continuous Random Variables –Joint Distribution (PDF) –Joint Density (pdf) –Marginal Densities –Independence 1

2. Multiple Random Variables In many experiments, the observations are expressible not as a single quantity, but as a family of quantities. For example to record the height and weight of each person in a community or the number of people and the total income in a family, we need two numbers. 2

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17. 17 Two Continuous Random Variables Let X and Y denote two random variables (r.v) based on a probability model. Then and

18. What about the probability that the pair of r.vs (X,Y) belongs to an arbitrary region D? In other words, how does one estimate, for example, Towards this, we define the joint probability distribution function of X and Y to be where x and y are arbitrary real numbers.

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37. we get (ii) To prove (1), we note that for and the mutually exclusive property of the events on the right side gives which proves (1). Similarly (2) follows. (1) (2) (i) Properties

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48. 48 It is useful to know the formula for differentiation under integrals. Let Then its derivative with respect to x is given by

49. The joint P.D.F and/or the joint p.d.f represent complete information about the r.vs, and their marginal p.d.fs can be evaluated from the joint p.d.f. However, given marginals, (most often) it will not be possible to compute the joint p.d.f. Consider the following example: Example : Given Obtain the marginal p.d.fs and Solution: It is given that the joint p.d.f is a constant in the shaded region in Fig.. To determine that constant c.

50. Thus c = 2. Using marginal pdf equations: and similarly Clearly, in this case given and above, it will not be possible to obtain the original joint p.d.f. Example : X and Y are said to be jointly normal (Gaussian) distributed, if their joint p.d.f has the following form: Eq:1

51. 51 Example of joint normal pdf rho=0.02 X,Y~N(0,1)

52. 52 By direct integration, and completing the square, it can be shown that ~ and similarly ~ Following the above notation, we will denote : (Eq:1) as Once again, knowing the marginals as above alone doesn’t tell us everything about the joint p.d.f in given in Eq#1 As we show below, the only situation where the marginal p.d.fs can be used to recover the joint p.d.f is when the random variables are statistically independent. PILLAI

53. Independence of r.vs Definition: The random variables X and Y are said to be statistically independent if the events and are independent events for any two sets A and B in x and y axes respectively. Applying the above definition to the events and we conclude that, if the r.vs X and Y are independent, then i.e., or equivalently, if X and Y are independent, then we must have Cond#1

54. If X and Y are discrete-type r.vs then their independence implies The previous equations give us the procedure to test for independence. Given: obtain the marginal p.d.fs and and examine whether Cond#1 is valid. If so, the r.vs are independent, otherwise they are dependent. Returning back to Previous Example, we observe by direct verification that Hence X and Y are dependent r.vs in that case.

55. 55 In case of two jointly Gaussian r.vs as in above are independent if and only if the fifth parameter is zero i.e.

56. Example : Given Determine whether X and Y are independent. Solution: Similarly In this case and hence X and Y are independent random variables.