Integral limits

  • #1
1,367
61

Main Question or Discussion Point

Hi.

I have the following expression

[tex]X=\frac{G\frac{x_1}{x_2}\gamma_Q}{\frac{1}{G}x_3 \gamma_P+1}[/tex]

where ##x_i## is an exponential random variable with mean 1. All other parameters are nonzero positive constants. Basically I want to find the probability density function (pdf) of ##X##. So I started with cumulative distribution function (cdf) as

[tex]F_X(x)=\text{Pr}\left[\frac{G\frac{x_1}{x_2}\gamma_Q}{\frac{1}{G}x_3 \gamma_P+1}\leq x\right][/tex]

which I evaluated it as

[tex]F_X(x)=\int_{x_3=0}^{\infty}\int_{x_3=0}^{\infty}\text{Pr}\left[x_1\leq\frac{x_2\,x}{G \gamma_Q}(\frac{1}{G}x_3 \gamma_P+1)\right]f_{X_2}(x_2)f_{X_3}(x_3)\,dx_2\,dx_3[/tex]

where ##f_{X_i}(x_i)## is the pdf of the random variable ##x_i##. Is what I did correct?

Thanks
 

Answers and Replies

  • #2
andrewkirk
Science Advisor
Homework Helper
Insights Author
Gold Member
3,792
1,390
It is only correct if the three random variables are independent. If they are, it is valid to factorise the joint pdf ##f_{X_1,X_2,X_3}(x_1,x_2,x_3)## into ##f_{X_1}(x_1)f_{X_2}(x_2)f_{X_3}(x_3)## as you have done. If they are not independent, that factorisation is incorrect.

Also, one of your two lower limits for integration needs to be ##x_2=0##.
 
  • #3
1,367
61
It is only correct if the three random variables are independent. If they are, it is valid to factorise the joint pdf ##f_{X_1,X_2,X_3}(x_1,x_2,x_3)## into ##f_{X_1}(x_1)f_{X_2}(x_2)f_{X_3}(x_3)## as you have done. If they are not independent, that factorisation is incorrect.

Also, one of your two lower limits for integration needs to be ##x_2=0##.
Yes, they are independent and identically distributed random variables. Right, the first integral is over all values of ##x_2##, it's a typo. Thanks
 
  • #4
andrewkirk
Science Advisor
Homework Helper
Insights Author
Gold Member
3,792
1,390
Yes, they are independent and identically distributed random variables. Right, the first integral is over all values of ##x_2##, it's a typo. Thanks
Given that additional info, and the discussed correction to the lower integration limit, what you've written is correct, but it is better practice to use upper case for random variables, to distinguish them from quantiles of those variables, ie:

$$F_X(x)=\int_{x_3=0}^{\infty}\int_{x_2=0}^{\infty}\text{Pr}\left[X_1\leq\frac{x_2\,x}{G \gamma_Q}(\frac{x_3 \gamma_P}{G}+1)\right]f_{X_2}(x_2)f_{X_3}(x_3)\,dx_2\,dx_3$$
 

Related Threads on Integral limits

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
4
Views
1K
Replies
7
Views
5K
Replies
3
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
12
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
Replies
7
Views
314
Top