Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Integral limits

  1. Apr 21, 2016 #1
    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
     
  2. jcsd
  3. Apr 21, 2016 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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##.
     
  4. Apr 21, 2016 #3
    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
     
  5. Apr 21, 2016 #4

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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:

    $$F_X(x)=\int_{x_3=0}^{\infty}\int_{x_2=0}^{\infty}\text{Pr}\left[X_1\leq\frac{x_2\,x}{G \gamma_Q}(\frac{x_3 \gamma_P}{G}+1)\right]f_{X_2}(x_2)f_{X_3}(x_3)\,dx_2\,dx_3$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Integral limits
  1. Limits of integrals (Replies: 9)

  2. Limit of an integral (Replies: 12)

  3. Limits and Integrals (Replies: 2)

  4. Limit of a integral (Replies: 5)

  5. Integration limits (Replies: 3)

Loading...