My Math Forum Joint Distribution help please

 April 22nd, 2012, 03:19 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: $f(x,y) = \left\{ \begin{array}{l l} x+y & \quad \text{if x\geq 0, y\leq 1}\\ 0 & \quad \text{otherwise}\\ \end{array} \right.$ 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, 12:50 PM   #2
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 Originally Posted by batman350z Can't get my head around this joint distribution problem. Suppose X and Y have joint distribution given by: $f(x,y) = \left\{ \begin{array}{l l} x+y & \quad \text{if x\geq 0, y\leq 1}\\ 0 & \quad \text{otherwise}\\ \end{array} \right.$ 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.
f(x,y) cannot be either a density function or a distribution function without further limits on x and y. y can be negative giving negative values for f for y < -x. x can be arbitrarily large - distribution would eventually be > 1.

 April 22nd, 2012, 01:02 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, $0 \leq x,y\leq1$, so I am guessing that would be interpreted as $0 \leq x \leq1, \ 0\leq y \leq 1$ If the above is the case, how would i go about it?
 April 23rd, 2012, 02:50 PM #4 Global Moderator   Joined: May 2007 Posts: 6,787 Thanks: 708 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 > t-1 and Y > t - X) so the integrals of f(x,y) is over t-1 < x < 1 and t-x < y < 1. For t < 1 the integral to get P(T < t) directly has 0 < x < t and 0 < y < t-x.

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