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So I don't really understand joint probability density functions. I've read that the joint density is the derivative of the joint probability function, but I don't understand what to do with the double integral of the density.

The function is $$f(y_1, y_2) = e^{-(y_1 + y_2)}$$ for $y_1>0, y_2>0$, 0 elsewhere.

By definition, the double integral of this thing across 0 to infinity is 1. So how does integrating this density get me to a distribution function?

Nick
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    The integral $\iint_D f(y_1,y_2),dy_1,dy_2$ gives you the probability that the random pair $(Y_1,Y_2)$ lands in region $D$. – André Nicolas Dec 13 '13 at 15:15
  • @AndréNicolas i've got that much- but that means we're talking specific values. how do I get from the density function to the distribution function? – Nick Dec 13 '13 at 15:25
  • You find the integral $\int_{-\infty}^{y_1}\int_{-\infty}^{y_2} f(t_1,t_2),dt_1,dt_2$, if you want the joint cdf $F_{Y_1,Y_2}(y_1,y_2)$. – André Nicolas Dec 13 '13 at 15:33
  • Ok, I got that part. so how would I go about determining an exact value for p(y1 + y2>3)? I don't see how to set up that integral. i know it'll be the complement of <3. – Nick Dec 13 '13 at 16:24
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    Draw the line $y_1+y_2=3$ (the $y_1$ axis can be what you usually call the $x$-axis, the $y_2$-axis the ordinary $y$-axis). You want the probability of being in the region $D$ above that line. This is the iterated integral $\int_{y_1=-\infty}^\infty \int_{y_2=3-y_1}^\infty f(y_1,y_2),dy_2,dy_1$. Of course you have to be careful about where the joint density is $0$. In the case of the one you gave, we integrate $e^{-(y_1+y_2)}$, $y_2$ from $3-y_1$ to $\infty$, and $y_1$ from $0$ to $\infty$. – André Nicolas Dec 13 '13 at 16:34

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