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Hi Guys,

Just need help understanding how to go about doing this question. I know how to convert single distributions but I'm unsure about how to do the joint ones. I usually draw a diagram representing the region I need to calculate but don't know to do next. Can anyone provide much needed assistnace?

Thanks in advance.

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1 Answer 1

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Think of the region bounded above by $x_2$ on the x-axis and above by $y_2$ on the y-axis. Your joint cumulative distribution function, $F_{X,Y}(x,y)$, is the probability $\mathbb{P}(X \le x, Y \le y)$. So you will want to subtract from that region the region bounded by $x_1, y_2$ and the region bounded by $x_2,y_1$. Since we doubled subtracted the region bounded by $x_1, y_1$, we need to add that back in.

Thus $\mathbb{P}(x_1 < X \le x_2, y_1 < Y \le y_2) = F_{X,Y}(x_2, y_2) - F_{X,Y}(x_1, y_2) - F_{X,Y}(x_2, y_1) + F_{X,Y}(x_1, y_1)$.

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How would you show this using general properties of distributions rather than a Venn-style type argument? –  AnonSubmitter85 May 27 '13 at 4:48
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