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I'm having trouble finding the a different order of integration.

Here is the problem:

The joint probability distribution of $X$ and $Y$, f($x,y$),is given by:

$\frac{6}{7}(x^2 + \frac{xy}{2})$, where $0 < x < 1, 0 < y < 2$.

I'm supposed to calculate $P(X>Y)$.

I $can$ calculate $P(X>Y)$, by doing this:

$$\int_0^1 \int_0^x \! f(x,y) \, \mathrm{d} y \mathrm{d} x,$$ where $f(x,y)$ is the joint probability distribution of X and Y mentioned above. (FWIW, the solution I get is 15/56.)

$Now$, $this$ $is$ $where$ $my$ $problem$ $lies$. If I would have happened to integrate dx first, instead of dy, I need to change my limits of integration. I would have changed them to:

$$\int_0^2 \int_y^1 \! f(x,y) \, \mathrm{d} x \mathrm{d} y$$

This does not give the right answer. Thus, my question is: how do I set up this integral?

ps. Yes, this is homework (which isn't graded, btw). But I have already solved it. I'm just trying to understand a different way of doing the problem.

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  • $\begingroup$ ZulfiqarIII has given a good answer. Put simply, in your second integral you need $y \le 1$ for the integral over $x$ and this needs to feed through into the limits of the integral over $y$. $\endgroup$
    – Henry
    Commented Jul 20, 2012 at 0:49

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The event $\{X > Y \}$ can be rewritten as the event $\{ (X,Y) \in B \}$, where $B := \{ (x,y) \in [0,1]\times [0,2] \; | \; x > y \}$; i.e. $$ P(X > Y) = P( (X,Y) \in B) = \int_B f(x,y) dx dy. $$ Now we can write this as an iterated integral, so we may choose an order of integration. First integrating over $y$ and then over $x$ indeed yields the first double integral you wrote down. However, if we first integrate over $x$ and then over $y$, we need to be a bit more careful. Note that $$ B = \{ (x,y) \in [0,1]\times [0,1] \; | \; x > y \} \cup \{ (x,y) \in [0,1]\times [1,2] \; | \; x > y \} = \{ (x,y) \in [0,1]\times [0,1] \; | \; x > y \} \cup \emptyset, $$ so we find that $$ P(X > Y) = \int_0^1 \int_y^1 f(x,y) dx dy. $$ Another way to get to this is by simply interchanging the order of integration in the first integral you wrote down (we may do so since the integral is non-negative), forgetting about $B$ entirely.

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