# Probability theory 2

1. Feb 6, 2008

### Milky

1. The problem statement, all variables and given/known data
The homework question I have is:

Let X be uniform over (0,1). Find E[X|X<1/2].

3. The attempt at a solution

First I found the density function: f(x|x<1/2] = f(x∩x<1/2) / f(x<1/2) = f(x<1/2) / f(x<1/2) = 1.

So, E[X|X<1/2] = ∫ xdx

My question is... are the limits of integration suppose to be from 0 to 1, or 0 to 1/2? Is everything else correct?

2. Feb 6, 2008

### EnumaElish

The limits of integration are {0, 1/2} because f(x|X<1/2) is defined (or is positive) for x<1/2 only.

This information should also alert you to the fact that $$\int_0^{1/2}f(x|X<1/2)dx = 1$$ has to be the case. Does f(x|x<1/2) = 1 satisfy this condition?

3. Feb 6, 2008

### Milky

Okay, new strategy. In my textbook in a similar problem, they did:

f(x|x<1/2) = f(x) / P{x<1/2}
When I do that, I get
f(x) / P{x<1/2} = 1 / (1/2) = 2.

E[x|x<1/2] = $$\int_0^{1/2}2xdx$$
= 1/4

Last edited: Feb 6, 2008