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Chapter 4. Multiple Random Variables

S. Chapter 4. Multiple Random Variables. In some random experiments, a number of different quantities are measured. Ex. 4.1. Select a student’s name from an urn. 4.1 Vector Random Variables.

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Chapter 4. Multiple Random Variables

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  1. S Chapter 4. Multiple Random Variables In some random experiments, a number of different quantities are measured. Ex. 4.1. Select a student’s name from an urn. tch-prob

  2. 4.1 Vector Random Variables A vector random variableX is a function that assigns a vector of real numbers to each outcome in S, the sample space of the random experiment. tch-prob

  3. Event Examples • Consider the two-dimensional random variable X=(X,Y). Find the region of the plane corresponding to events tch-prob

  4. Product Form • We are particularly interested in events that have the product form y2 y1 x1 x2 tch-prob

  5. Product Form • A fundamental problem in modeling a system with a vector random variable involves specifying the probability of product-form events • Many events of interest are not of product form. • However, the non-product-form events can be approximated by the union of product-form events. Ex. tch-prob

  6. 4.2 Pairs of Random variables A. Pairs of discrete random variables - Let X=(X,Y) assume values from - The joint pmf of X is It gives the probability of the occurrence of the pair - The probability of any event A is the sum of the pmf over the outcomes in A: - When A=S, tch-prob

  7. Marginal pmf • We are also interested in the probabilities of events involving each of the random variables in isolation. • These can be found in terms of the Marginal pmf. • In general, knowledge of the marginal pmf’s is insufficient to specify the joint pmf. tch-prob

  8. Ex. 4.6. Loaded dice: A random experiment consists of tossing two loaded dice and noting the pair of numbers (X,Y) facing up. The joint pmf k j The marginal pmf P[X=j]=P[Y=k]=1/6. tch-prob

  9. Ex. 4.7. Packetization problem: The number of bytes N in a message has a geometric distribution with parameter 1-p and range SN={0,1,2,….}. Suppose that messages are broken into packets of maximum length M bytes.Let Q be the number of full packets and let R be the number of bytes left over. Find the joint pmf and marginal pmf’s of Q and R. tch-prob

  10. joint cdf of X and Y The joint cdf of X and Y is defined as the probability of the product-form event marginal cdf tch-prob

  11. joint cdf of X and Y tch-prob

  12. A B B tch-prob

  13. joint pdf of two jointly continuous random variables X and Y are jointly Continuous if the probabilities of events involving (X,Y) can be expressed as an integral of a pdf, . tch-prob

  14. Marginal pdf: obtained by integrating out the variables that are not of interest. tch-prob

  15. Ex. 4.10. A randomly selected point (X,Y) in the unit square has uniform joint pdf given by tch-prob

  16. Ex. 4.11 Find the normalization constant c and the marginal pdf’s for the following joint pdf: tch-prob

  17. Ex. 4.12 0 1 tch-prob

  18. Ex. 4.13 The joint pdf of X and Y is We say that X and Y are jointly Gaussian. Find the marginal pdf’s. tch-prob

  19. 4.3 Independence of Two Random Variables X and Y are independent random variables if any event A1 defined in terms of X is independent of any event A2 defined in terms of Y; P[ X in A1, Y in A2 ] = P[ X in A1 ] P[ Y in A2 ] Suppose that X,Y are discrete random variables, and suppose we are interested in the probability of the event where A1 involves only X and A2 involves only Y. “”If X and Y are independent, then A1 and A2 are independent events. Let tch-prob

  20. “” tch-prob

  21. In general, X, Y are independent iff If X and Y are independent r.v. ,then g(X) and h(Y) are also independent. # A and A’ are equivalent events; B and B’ are equivalent events. tch-prob

  22. Ex.4.15 In the loaded dice experiment in Ex. 4.6, the tosses are not independent. Ex. 4.16 Q and R in Ex. 4.7 are independent. Ex.4.17 X and Y in Ex. 4.11 are not independent, even though the joint pdf appears to factor. tch-prob

  23. 4.4 Conditional Probability and Conditional Expectation Many random variables of practical interest are not independent. We are interested in the probability P[Y in A] given X=x? conditional probability A. If X is discrete, can obtain conditional cdf of Y given X=xk The conditional pdf, if the derivative exists, is tch-prob

  24. If X and Y are independent - If X and Y are discrete If X and Y are independent tch-prob

  25. B. If X is continuous, P[ X = x] = 0 conditional cdf of Y given X = x conditional pdf. tch-prob

  26. Discrete continuous discrete continuous Theorem on total probability tch-prob

  27. tch-prob

  28. tch-prob

  29. tch-prob

  30. Conditional Expectation The conditional expectation of Y given X=x is or if X,Y are discrete. tch-prob

  31. can be generalized to tch-prob

  32. [ X Y ] [ 0,0 ] 0.1 [ 1,0 ] [ 1,1 ] [ 2,0 ] [ 2,1 ] [ 2,2 ] [ 3,0 ] [ 3,1 ] [ 3,2 ] [ 3,3 ] E[Y] = 1 E[X] = 2.0 tch-prob

  33. tch-prob

  34. Ex. 4.25 Find the mean of Y in Ex. 4.22 using conditional expectation. Ex. 4.26 Find the mean and variance of the number of customer arrivals N during the service time T of a specific customer in Ex. 4.23. tch-prob

  35. 4.5 Multiple Random Variables Extend the methods for specifying probabilities of pairs of random variables to the case of n random variables. We say that are jointly continuous random variables if tch-prob

  36. tch-prob

  37. X1 and X3 are independent zero-mean, unit-variance Gaussian r.v.s. tch-prob

  38. Independence tch-prob

  39. 4.6 Functions of Several Random Variables Quite often we are interested in one or more functions of random variables involved with some experiment. For example, sum, maximum or minimum of X1, X2, …,Xn. tch-prob

  40. Example 4.31 Z=X+Y Superposition integral If X and Y are independent r.v., convolution integral tch-prob

  41. Example 4.32 Sum of Non-Independent r.v.s Z=X+Y , X,Y zero-mean, unit-variance with correlation coefficient tch-prob

  42. Sum of these two non-independent Gaussian r.v.s is also a Gaussian r.v. tch-prob

  43. Ex.4.33 A system with standby redundancy. Let T1 and T2 be the lifetimes of the two components. They are independent exponentially distributed with the same mean. The system lifetime is Erlang m=2 tch-prob

  44. The conditional pdf can be used to find the pdf of a function of several random variables. Let Z = g (X,Y). Given Y = y, Z = g (X,y) is a function of one r.v. X. Can first find from then find tch-prob

  45. Example 4.34 Z = X/Y X,Y indep., exponentially distributed with mean one. Assume Y = y, Z = X/y is a scaled version of X tch-prob

  46. tch-prob

  47. tch-prob

  48. (z, z) tch-prob

  49. (z, z) tch-prob

  50. Transformation of Random Vectors Joint cdf of tch-prob

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