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MATH 3033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Slides by Michael Maurizi Format by Tim Birbeck Instructor Longin Jan Latecki C9: Joint Distributions and Independence 9.1 – Joint Distributions of Discrete Random Variables Joint Distribution: the combined distribution of two or more random variables defined on the same sample space Ω Joint Distribution of two discrete random variables: The joint distribution of two discrete random variables X and Y can be obtained by using the probabilities of all possible values of the pair (X,Y) Joint Probability Mass function p of two discrete random variables X and Y: p : R 2 [0,1] p XY (a, b) p(a, b) P(X a, Y b) for a, b Joint Distribution function F of two random variables X and Y: Can be thought of as the sum of the elements in box it makes with the upper-left corner. p : R 2 [0,1] F (a, b) P(X a, Y b) for a, b 9.1 – Joint Distributions of Discrete Random Variables Marginal Distribution: Obtained by adding up the rows or columns of a joint probability mass function table. Literally written in the margins. Example: Joint Distribution of S and M. S = The sum of two dice, M = The maximum of two dice. b pS(a) a 1 2 3 4 5 6 2 1/36 0 0 0 0 0 1/36 3 0 2/36 0 0 0 0 2/36 4 0 1/36 2/36 0 0 0 3/36 5 0 0 2/36 2/36 0 0 4/36 6 0 0 1/36 2/36 2/36 0 5/36 7 0 0 0 2/36 2/36 2/36 6/36 Marginal distribution function of Y: 8 0 0 0 1/36 2/36 2/36 5/36 FY (b) P(Y b) F (b,) lim F (a, b) 9 0 0 0 0 2/36 2/36 4/36 10 0 0 0 0 1/36 2/36 3/36 11 0 0 0 0 0 2/36 2/36 12 0 0 0 0 0 1/36 1/36 pM(a) 1/36 3/36 5/36 7/36 9/36 11/36 1 Marginal distribution function of X: FX (a) P(X a) F (a,) lim F (a, b) b a 9.2 – Joint Distributions of Continuous Random Variables Joint Continuous Distribution: Like an ordinary continuous random variable, only works for a range of values. There must exist a function f that fulfills the following properties for there to be a joint continuous distribution: 2 f :R R b1 b2 P(a1 X b1 , a2 Y b2 ) f ( x, y )dxdy for all a1 b1 and a2 b2 a1 a2 f ( x, y ) 0 for all x and y f ( x, y)dxdy 1 9.2 – Joint Distributions of Continuous Random Variables Joint distribution function: F(a,b) can be constructed given f(x,y), and vice versa a b F ( a, b) f ( x, y)dxdy and 2 f(x,y) F ( x, y ) xy Marginal probability density function: You need to integrate out the unwanted random variable to get the marginal distribution. f X ( x) f ( x, y)dy and fY ( y ) f ( x, y)dx 9.3 – More than Two Random Variables Assuming we have n random variables X1, X2, X3, … Xn. We can get the joint distribution function and the joint probability mass functions. for a1 , a2 , , an Joint distributi on function : F (a1 , a2 , , an ) P(X1 a1 , X 2 a2 , , X n an ) Joint probabilit y mass function : p(a1 , a2 , , an ) P(X1 a1 , X 2 a2 , , X n an ) 9.4 – Independent Random Variables Tests for Independence: Two random variables X and Y are independent if and only if every event involving X is independent of every event involving Y. This also applies to joint distributions using more than two random variables. for all possible a and b P(X a, Y b) P(X a ) P(Y b) or P(X a, Y b) P(X a ) P(Y b) or F (a, b) FX (a ) FY (b) 9.5 – Propagation of Independence Independence after a change of variable: If a function is applied to several independent random variables, the new resulting random variables will also be independent.