1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Probability Mass Function of { Z | Z < 1 }, Z given Z is less than 1

  1. May 15, 2010 #1
    Probability Mass Function of { Z | Z < 1 }, "Z given Z is less than 1"

    1. The problem statement, all variables and given/known data

    Given Z = X + Y.

    Find the probability density function of Z|Z < 1.


    2. Relevant equations

    N.A.

    3. The attempt at a solution

    f(z) = P(Z=z|Z<1) = P(Z=z AND Z < 1) / P(Z < 1).

    I thought the top could be simplified to P(Z=z) for z < 1. Correct?

    So,

    f(z) = 0, for z > or = 1
    f(z) = P(Z=z)/P(Z<1), for z < 1.

    Correct?
     
  2. jcsd
  3. May 15, 2010 #2
    Re: Probability Mass Function of { Z | Z < 1 }, "Z given Z is less than 1"

    What you have is essentially correct, although it seems bad form to use probabilities in the statement of a probability density function.

    I think better is to write: Let g(z) be the PDF of Z = X + Y. If this is still the problem with Z = X + Y where X and Y are uniform on [0,1] that you've been working on in different ways, then you have this. But even if it's not you can just write it in terms of g(z).

    So then:

    [tex] f(z)=\left\{\begin{array}{cc}0,&\mbox{ if }
    z > 1\\ \frac{g(z)}{\int_{-\infty}^1 g(z) dz}, & \mbox{ if } z \leq 1\end{array}\right [/tex]
     
    Last edited: May 15, 2010
  4. May 16, 2010 #3
    Re: Probability Mass Function of { Z | Z < 1 }, "Z given Z is less than 1"

    hgfalling, thank you so much for all the probability help!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook