The joint density function of Y1 and Y2 is given as f(y1, y2) = e^-y1, for 0 <= y2 <= y1 < infin

First, we need to find the probability P(Y1 < 2Y2 > 1). To do this, we will integrate the joint

The joint density function of Y1 and Y2 is given as f(y1,...

The joint density function of Y1 and Y2 is given as f(y1, y2) = e^-y1, for 0 <= y2 <= y1 < infinity; and 0 otherwise. (a) Find P(Y1 < 2, Y2 > 1). (b) Find P(Y1 >= 2Y2). (c) Find the marginal density functions for Y1 and Y2.

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Transcript

00:01 So if you look at this, what they're really asking for is the double integral of that's your function right there and then they give you your bounds.

00:10 So the easiest way to start with this is to try to find your region right here, your r, and we need to graph out these bounds.

00:21 So those are your y graphs.

00:24 So the drawing can be as rough as it needs to be, but negative 2x plus 4 is gonna look something like that.

00:32 Negative 2x plus 7 is gonna look like that, roughly.

00:40 And that's all it needs to be.

00:42 And then y equals x minus 2, which is like here.

00:54 Y equals x plus 1.

00:59 So that's something like there.

01:03 So you want that region right there, the one that is colored in.

01:08 So for this, so if we look at these bounds, we see that the x bound starts here, that's the leftmost on the x, and it goes over to here.

01:23 And then the y starts down there and it goes up to there, to this point.

01:31 So we can start writing that as an inequality.

01:35 We'll say x starts at 2, that's where that orange dot is.

01:39 So x goes from 2 and then this intersection right here is at x equals 3.

01:49 And then the y bounds start with this lower line and it goes up to this top one.

01:55 So that's x minus 2 to x plus 1.

01:59 So x minus 2, y, and then x plus 1...

Question

The joint density function of Y1 and Y2 is given as f(y1, y2) = e^-y1, for 0 <= y2 <= y1 < infinity; and 0 otherwise. (a) Find P(Y1 < 2, Y2 > 1). (b) Find P(Y1 >= 2Y2). (c) Find the marginal density functions for Y1 and Y2.

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