Randomly truncated pdf

1. Jul 19, 2009

benjaminmar8

Hi, all,

I am having a problem in calculating a randomly truncated pdf. Let x be a random variable, it's pdf f(x) is known. Let t1 and t2 be anther two random variables, their pdf f(t1) and f(t2) are known as well. Now, how do I compute the pdf f(x|t1<x<t2)?

Thks a lot.

2. Jul 19, 2009

John Creighto

$$f(x|t_1<x<t_2)=\int_{-\infty}^{-\infty}\int_{-\infty}^{-\infty}f(x)rect(x,t_1,t_2)f(t_1)f(t_2)dt_1dt_2$$

where $$rect(x,t_1,t_2)$$ is defined to be $$1$$ if $$t_1<x<t_2$$ and $$0$$ otherwise.

3. Jul 19, 2009

benjaminmar8

But the question is, how do I know when t1<X<t2 since t1 and t2 are random?

4. Jul 19, 2009

John Creighto

You don't. You consider all possibles for t1, and t2 and the probability of each possibility.

5. Jul 20, 2009

benjaminmar8

I did a couple of simulations and found that the pdf f(x|t1<x<t2) seems need to be scaled. Maybe I have miss out some conditions, say the support of x, t1 and t2 are all [0,R]. In this case, how do I compute the truncated pdf? Thanks a lot.

6. Jul 20, 2009

John Creighto

I'm sorry. What I gave you wasn't really f(x|t1<x<t2). To get the conventional probability, simply devide f(x) by the integral of f(x) from t1 to t2. However, the contional probability is not the same thing as a randomly truncated PDF. What I gave you is the distribution of f(x) given some random truncation. I'm not sure which you want because I don't know much about the problem you are trying to solve.

7. Jul 22, 2009

benjaminmar8

what I am trying to solve is the desnity function of f(x|t1<x<t2), therefore, its intergral over the support should be 1. What you gave me seems should be devided by 1/(F(t2)-F(t1)) (and you mentioned that), however, since t2 and t1 are random, I use its expectation instead. That's to say, the scaling is 1/(F(E[t2])-F(E[t1])). I know this is an approximation, how do I compute it in an exact manner? Thank u very much.

8. Aug 4, 2009