Results 1 to 2 of 2

Math Help - Independence of events

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    10

    Independence of events

    Let X and Y be random variables that both have an exponential distribution with parameters lambda and mu respectively. X and Y are independent.

    Define Z = min{X, Y}

    (1) Prove that the event {X < Y} is independent of the event {Z > z}

    (2) Find E[ e^(-z) | X ].
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    You can calculate the three probabilities.
    That's probably not a clever way, but it must work.

    P(X<Y,\min\{X,Y\}>z)=P(X<Y,X>z)

    ={1\over \mu\lambda}\int_z^{\infty}\int_z^y e^{-x/\lambda}e^{-y/\mu}dxdy

    Then get

    P(X<Y)={1\over \mu\lambda}\int_0^{\infty}\int_0^y e^{-x/\lambda}e^{-y/\mu}dxdy

    and

    P(\min\{X,Y\}>z)= P(X>z,Y>z)= P(X>z)P(Y>z)

    ={1\over \mu\lambda}\int_z^{\infty}e^{-x/\lambda}dx\int_z^y e^{-y/\mu}dy=e^{-z/\lambda}e^{-z/\mu}

    Then compare to see if P(AB)=P(A)P(B).
    Last edited by matheagle; August 28th 2009 at 11:31 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: November 15th 2010, 01:24 PM
  2. events
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: July 15th 2008, 02:41 AM
  3. events
    Posted in the Statistics Forum
    Replies: 1
    Last Post: November 4th 2007, 01:11 PM
  4. events
    Posted in the Statistics Forum
    Replies: 1
    Last Post: October 31st 2007, 03:20 PM
  5. Help two events
    Posted in the Statistics Forum
    Replies: 2
    Last Post: October 20th 2007, 03:41 PM

Search Tags


/mathhelpforum @mathhelpforum