ESM3A HW9 - Jacobs University Bremen School of Engineering...

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Jacobs University Bremen School of Engineering and Science Peter Oswald Fall Term 2010 120201 ESM3A — Problem Set 9 Issued: 10.11.2010 Due: Wednesday 17.11.2010 (in class) This problem set is about several random variables (joint distributions, independence, con- ditioning). 9.1. Random variables X and Y are said to be jointly continuous if there is a function f X,Y : IR 2 [0 , ), called the joint density , such that P ( X ( a,b Y ( c,d ]) = Z d c Z b a f X,Y ( x,y )d x d y for every a b and c d . (a) Show that jointly continuous random variables are independent if and only if we have f X,Y ( x,y ) = f X ( x ) f Y ( y ) for all such x,y IR, where these functions are continuous. (b) Give an example of two jointly continuous random variables that are not independent. 9.2. Let the joint probability density function of X and Y be given by f X,Y ( t,s ) = 12 ts 2 , 0 s t 1 , f X,Y ( t,s ) = 0 otherwise . (a) Determine if
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