(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

$$L=\left\{\begin{array}{cc}X,&\mbox{ if } Y\leq X\\1-X, & \mbox{ if } Y>Z\end{array}\right$$

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?

$$E_0(L) = \int_{0}^{1} \int_{0}^{x} x \, dy \,dx$$

$$E_1(L) = \int_{0}^{1} \int_{x}^{1} 1 - x \, dy \,dx$$

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