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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 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 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 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 at 19:10

1 Answer 1

Since you have joint probability distribution then you have to use the double integral. The first integral will be refered to the first random variable and the second to the second one, i.e.:

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

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