Calculating Randomly Truncated PDF for X given T1 < X < T2

In summary: F[x|t1<x<t2] = P[t1<X<t2 & X<=x] / P[t1<X<t2]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])).
  • #1
benjaminmar8
10
0
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.
 
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  • #2
[tex]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[/tex]

where [tex]rect(x,t_1,t_2)[/tex] is defined to be [tex]1[/tex] if [tex]t_1<x<t_2[/tex] and [tex]0[/tex] otherwise.
 
  • #3
John Creighto said:
[tex]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[/tex]

where [tex]rect(x,t_1,t_2)[/tex] is defined to be [tex]1[/tex] if [tex]t_1<x<t_2[/tex] and [tex]0[/tex] otherwise.

But the question is, how do I know when t1<X<t2 since t1 and t2 are random?
 
  • #4
benjaminmar8 said:
But the question is, how do I know when t1<X<t2 since t1 and t2 are random?

You don't. You consider all possibles for t1, and t2 and the probability of each possibility.
 
  • #5
John Creighto said:
[tex]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[/tex]

where [tex]rect(x,t_1,t_2)[/tex] is defined to be [tex]1[/tex] if [tex]t_1<x<t_2[/tex] and [tex]0[/tex] otherwise.

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
benjaminmar8 said:
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.

I'm sorry. What I gave you wasn't really f(x|t1<x<t2). To get the conventional probability, simply divide 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
John Creighto said:
I'm sorry. What I gave you wasn't really f(x|t1<x<t2). To get the conventional probability, simply divide 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.

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
I'd start with the CDF and differentiate.

F[x|t1<x<t2] = P[t1<X<t2 & X<=x] / P[t1<X<t2]

both those probabilities can be written as integrals of functions of the pdf's.
 

1. What is a randomly truncated PDF?

A randomly truncated PDF is a probability distribution function that has been cut off at a certain point, resulting in a truncated version of the original distribution. This means that the values beyond the truncation point are no longer included in the distribution.

2. How is a randomly truncated PDF different from a regular PDF?

A randomly truncated PDF differs from a regular PDF in that it has a limited range of values due to the truncation. This can affect the shape and characteristics of the distribution, such as the mean and variance.

3. What are the applications of randomly truncated PDF?

Randomly truncated PDFs are commonly used in statistical analysis to model situations where a certain outcome is impossible or not relevant. They can also be used in finance and economics to model skewed data, as well as in risk management and insurance.

4. How is a randomly truncated PDF calculated?

A randomly truncated PDF is calculated by first determining the desired truncation point. Then, the original PDF is integrated from the truncation point to infinity. This value is then subtracted from the original PDF to create the truncated version.

5. What are some common examples of randomly truncated PDFs?

Some common examples of randomly truncated PDFs include the truncated normal distribution, which is used to model data that is limited to a certain range, and the truncated exponential distribution, which is commonly used in reliability analysis to account for situations where an event cannot occur beyond a certain time. Other examples include the truncated Poisson, binomial, and gamma distributions.

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