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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!
     
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