# Statistics: Integration from a joint probability distribution

If the joint probability density of two random variables is given by: $$f(x_1, x_2) = \begin{cases}6e^{-2x_1-3x_2} &\quad \text{for } x_1 > 0,\, x_2 > 0\\ 0,&\quad \text{elsewhere}.\end{cases}$$

Find the probabilities that:

(a) the first random variable will take on a value between $1$ and $2$ and the second random variable will take on a value between $2$ and $3$;

(b) the first random variable will take on a value less than $2$ and the second random variable will take on a value greater than $2$.

The solution gives a formula with two integral symbols with the bound of $1,2$ and $2,3$ respectively. I've never seen this before. Is this another way to represent integration by parts? Can I get some help with how to do this?

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"I've never seen this before." How would YOU write it down, based on the similar exercises you previously solved? – Did Feb 27 '14 at 18:43
@Did If by similar exercises you mean other joint distributions, we just started this topic in class. But I still want to know if this "double integration" thing is the same as integration by parts, because I have done that in previous classes. – UnworthyToast Feb 27 '14 at 18:55
No this is not the same. Actually there is no integration by parts in the picture, but iterated integrals. – Did Feb 27 '14 at 18:59
@Did Ok I'm doing some research on double integration now and it doesn't seem too bad. Thanks to all for giving me a shove in the right direction! – UnworthyToast Feb 27 '14 at 19:10

a)$$P(1<X_1<2,2<X_2<3)=\int^{2}_{1}\int^{3}_{2} 6e^{-2x_1-3x_2}dx_1dx_2$$
b)$$P(X_1<2,X_2>2)=\int^{2}_{0}\int^{\infty}_{2} 6e^{-2x_1-3x_2}dx_1dx_2$$