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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Prof. Dr. Peter Oswald
Probability distribution, Probability theory, probability density function, Jacobs University Bremen School of Engineering