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April 22nd, 2012, 04:19 AM  #1 
Newbie Joined: Apr 2012 From: Canada Posts: 7 Thanks: 0  Joint Distribution help please
Can't get my head around this joint distribution problem. Suppose X and Y have joint distribution given by: Find the distribution of X+Y Having difficulties finding the boundaries for my integration. I know that I need do declare a dummy variable t = X + Y Been stuck on this for a few hours now, help is much appreciated! Thanks. 
April 22nd, 2012, 01:50 PM  #2  
Global Moderator Joined: May 2007 Posts: 6,856 Thanks: 745  Re: Joint Distribution help please Quote:
 
April 22nd, 2012, 02:02 PM  #3 
Newbie Joined: Apr 2012 From: Canada Posts: 7 Thanks: 0  Re: Joint Distribution help please
Hmm, I might be misunderstanding the boundaries. The question actually has them written as, , so I am guessing that would be interpreted as If the above is the case, how would i go about it? 
April 23rd, 2012, 03:50 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,856 Thanks: 745  Re: Joint Distribution help please
First f(x,y) is a density function, not a distribution function. Let T = X + Y. What you want to calculate is P(T < t) = P(X+Y < t) = P(X < t and Y < t  X). To do this simply integrate the density function over the domain specified, 0 < x < t, 0 < y < t  x. The main thing you have to be careful of is the fact that f(x,y) = 0 outside the unit square (in x,y), so that you have to break up the integration into the parts where f(x,y) = x + y and f(x,y) = 0. For 0 < t < 1, there is no problem, but for 1 < t < 2 you need to do the breakup. For t > 2, the probability = 1. Post script: for t > 1, it is easier to work with P(T > t) and the use P(T < t) = 1  P(T > t). In this case P(T > t) = P(X > t1 and Y > t  X) so the integrals of f(x,y) is over t1 < x < 1 and tx < y < 1. For t < 1 the integral to get P(T < t) directly has 0 < x < t and 0 < y < tx. 

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