# Conditional distribution for random variable on interval

1. Feb 4, 2014

### Srumix

1. The problem statement, all variables and given/known data

Find the conditional distribution function and density for the random variable X defined on R given that X is in some interval I = (a,b) where P(X in I) > 0. Assume that the density and distribution for the random variable X is known

2. Relevant equations

fX|X$\in$I = P(X$\leq$x | X$\in$ I) = fX,X$\in$I(x,x)/fX$\in$I(x)

3. The attempt at a solution

I'm sorry, but my latex skills are very poor so I will try to describe in words what my problem is.
The problem I'm having is that I know how to calculate the probability of P(X in I) since we just take the integral of the density function over the interval I in question. However, what do I do with the "joint" distribution fX,X$\in$I(x,x) that I need for the definition of conditional distribution? That is what I can't figure out.

The conditional density is the coefficient of $\Delta x$ in the first-order (small-$\Delta x$) expansion of
$$\text{P} \{ x < X < x + \Delta x | a \leq X \leq b \} = \frac{\text{P} \{ x < X < x+ \Delta x \: \& \: a \leq X \leq b \}}{\text{P} \{ a \leq X \leq b \}}$$
For $x \in (a,b)$, can you figure out what is the numerator, in terms of the probability density function $f(.)$? Can you figure out the denominator?