(Easy?) Probability Q: Uniform in Uniform

[ryzeg]

Hello

I have been struggling with a simple probability question.

Homework Statement

We are given that X is a uniformly distributed random variable on [0, 1]. After X is chosen, we take another uniform [0, 1] random variable Y (independent of X) and choose the subinterval that Y falls in. L is the length of this chosen subinterval (either [0, X] or (X, 1]), and we want its expectation.

Homework Equations

The expectation of a uniform random variable is (a+b)/2, where a and b are the interval bounds. These equations may also be useful:

f_{Y}(y|X=x) = \frac{f(x,y)}{f_{X}(x)}

f_{X}(x|Y=y) = \frac{f_{Y}(y|X=x)f_{X}(x)}{f_{Y}(y)}

E(g(Y)|X=x) = \int \! g(y)f_{Y}(y|X=x) \,dx

E(Y) = \int \! E(Y|X=x)f_{X}(x) \,dx

The Attempt at a Solution

I notice

L=\left\{\begin{array}{cc}X,&amp;\mbox{ if }<br /> Y\leq X\\1-X, &amp; \mbox{ if } Y&gt;Z\end{array}\right

And E(X) is obviously 1/2... but I do not know how to find E(L) :(

I tried doing double integrals for expectation on both cases, and summing these, but I get 1/2 which doesn't seem right...

E_0(L) = \int_{0}^{1} \int_{0}^{x} \! 1 y \, dy \,dx = 1/3

E_1(L) = \int_{0}^{1} \int_{x}^{1} \! 1 y \, dy \,dx = 1/6

E(L) = E_0 + E_1 = 1/3 + 1/6 = 1/2

Any help?

 

[hgfalling]

Where is the "y" in your expectation integrals coming from?
The value function is either f(x) = x (when y < x) or f(x) = 1 - x (when y > x). I don't see that in your integrals.

Recall that \int f(x)g(x) dx is the integral for expectation where f(x) is the value function and g(x) is the density. Make sure when you construct your integrals that you have the right values and densities.

 

[ryzeg]

I was using y as the value function, with the bounds ranging over all possible values.

When I do integrals with x*1 from 0 to 1, I still get 1/2.

 

[hgfalling]

Are these your integrals?

<br /> E_0(L) = \int_{0}^{1} \int_{0}^{x} x \, dy \,dx <br />

<br /> E_1(L) = \int_{0}^{1} \int_{x}^{1} 1 - x \, dy \,dx <br />

They don't add up to 1/2.

 

[ryzeg]

Yeah, you're right, I did my integrals wrong. By symmetry, it is obvious that it should be 2/3, which is what those add up to. Thanks!

 

[mults]

hi there I am new with this stuff but need some help... I've given a question in Finding C,D, F(x). Where the mean is 10 standard deviation is 1. (Uniform Probability distribution) Can you tell me which formula i should used...
Thanks