Results 1 to 4 of 4

Math Help - An exponential random variable question

  1. #1
    Newbie
    Joined
    Dec 2012
    From
    Israel
    Posts
    2

    An exponential random variable question

    Hello to all,

    let's say i have 2 i.i.d exponential random variable x1 and x2 with parameter m.

    i want to find out the probability that x.1 is equal to x.2 : P(x.1 = x.2)

    so what i did is to say that by conditioning on x.2 i get : (1) P(x.1 = x.2) = P(x.1 = k | x.2 = k)*P(x.2 = k) and i will sum over all possible values of k (by integral) [0,Infinity].

    because we have 2 i.i.d variables i can say that : (2) P(x.1 = k | x.2 = k) = P(x.1 = k)

    so by inserting equation (1) -> (2) i get (3) P(x.1 = x.2) = P(x.1 = k)*P(x.2 = k) sum over all possible values of k (by integral) [0,Infinity].

    now all i have to do is to place P(x.1 = k) and P(x.2 = k) into (3) and i know that P(x = k) = f(k) = m*e^(-m*k) for both variables because they are i.i.d.

    so i get equation no. (4) P(x.1 = x.2) = (m^2)*e^(-2*m*k) sum over all possible values of k (by integral) [0,Infinity].

    if i solve this integral i get (5) P(x.1 = x.2) = m/2

    everything seems o.k by the way i developed this answer but what if m=5 ? the probability can't be 2.5 but m surely can be 5...

    so where did i get it wrong ? in the way or in the understanding of the answer i got ?

    thanks in advance...


    Attachment 26096
    Attached Thumbnails Attached Thumbnails An exponential random variable question-untitled.png  
    Last edited by DudeM; December 5th 2012 at 01:37 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,881
    Thanks
    697

    Re: An exponential random variable question

    Hey DudeM.

    You won't be able to check an exact inequality since it is a continuous random variable, but what you can do is check an interval.

    So basically you want to find whether the difference lies in a particular interval of [-e,e] for some small e > 0.

    You can find the sum of two variables distribution by using convolution.

    From this:

    List of convolutions of probability distributions - Wikipedia, the free encyclopedia

    we can see that the distribution will be a gamma however this is only for summing random variables. Yheoutiou will need to find the PDF of the -X where X is exponential and then use the convolution theorem to get the PDF.

    Once you have the PDF then just calculate the probability in [-e,e]
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2012
    From
    Israel
    Posts
    2

    Re: An exponential random variable question

    "You won't be able to check an exact inequality since it is a continuous random variable..."

    what do you mean by that? by using the exponential variable pdf i can find the probability that x.1 = k , since P(x.1 = k) = f.x(k) -> the pdf at point k.

    by the way , why did you say i was summing random variable.... maybe i do x.1 - x.2, ( from P(x.1 = x.2) ) thats alot different...
    Last edited by DudeM; December 5th 2012 at 10:05 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,881
    Thanks
    697

    Re: An exponential random variable question

    What I mean is that for any continuous random variable P(X = x) is 0.

    This is not the same as the probability density function which has a specific non-zero, valid probability value.

    The actual probability of getting one observation as opposed to a range of probabilities is zero.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Exponential Random Variable Problem
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 2nd 2012, 02:29 PM
  2. Exponential random variable
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: March 19th 2011, 09:10 PM
  3. Exponential random variable: prove μ=σ
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: October 21st 2010, 05:32 PM
  4. exponential random variable with a random mean?
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: March 21st 2010, 02:05 PM
  5. exponential random variable
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 17th 2009, 06:54 AM

Search Tags


/mathhelpforum @mathhelpforum