Conditional distribution for random variable on interval

[Srumix]

Homework Statement

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

Homework Equations

f_{X|X[itex]\in[/itex]I} = P(X[itex]\leq[/itex]x | X[itex]\in[/itex] I) = f_{X,X[itex]\in[/itex]I}(x,x)/f_{X[itex]\in[/itex]I}(x)

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 f_{X,X[itex]\in[/itex]I}(x,x) that I need for the definition of conditional distribution? That is what I can't figure out.

Thanks in advance!

 

[Ray Vickson]

Homework Statement

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

Homework Equations

f_{X|X[itex]\in[/itex]I} = P(X[itex]\leq[/itex]x | X[itex]\in[/itex] I) = f_{X,X[itex]\in[/itex]I}(x,x)/f_{X[itex]\in[/itex]I}(x)

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 f_{X,X[itex]\in[/itex]I}(x,x) that I need for the definition of conditional distribution? That is what I can't figure out.

Thanks in advance!

The conditional density is the coefficient of ##\Delta x## in the first-order (small-##\Delta x##) expansion of
[tex]\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 \}} [/tex]
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?