Finding probability given joint pdf.

[cookiesyum]

Homework Statement

Let X1 X2 X3 be independent and identically distributed random variables with common pdf f(x) = e^-x 0<x<infinity, zero elsewhere.

Evaluate P(X1 < X2|X1 < 2X2) and P(X1 < X2 < X3|X3 < 1).

The Attempt at a Solution

Would appreciate any hints as to where to start...

 

[clamtrox]

I haven't actually studied any statistics so there might be an easier way of doing this but this is how I'd do it...

Having a conditional probability means that you essentially have a 100% probability for the condition, that is

$$\int_{x_1 < 2x_2} dx_1 dx_2 f(x_1, x_2) = 1$$

This gives you the probability density given the condition that x_1 < 2 x_2. You can then use that to find the probability that x_1 < x_2.

 

[cookiesyum]

I haven't actually studied any statistics so there might be an easier way of doing this but this is how I'd do it...

Having a conditional probability means that you essentially have a 100% probability for the condition, that is

$$\int_{x_1 < 2x_2} dx_1 dx_2 f(x_1, x_2) = 1$$

This gives you the probability density given the condition that x_1 < 2 x_2. You can then use that to find the probability that x_1 < x_2.

But I don't think the problem is saying X1 < 2X2 I think the phrasing was X2|X1 < 2X2. Am I thinking about it wrong?

 

[clamtrox]

The first problem reads out as "probability that x_1 < x_2, given that you already know that x_1 < 2x_2". The 2nd problem is "probability that x_1 < x_2 < x_3, given that you know that x_3 < 1". Atleast that is what this notation means as far as I know. :-)