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Normal random variables

  1. Mar 29, 2009 #1
    1. The problem statement, all variables and given/known data
    Let [tex]\psi[/tex](x) = 2[tex]\phi[/tex](x) - 1. The function [tex]\psi[/tex] is called the positive normal distribution. Prove that if Z is standard normal, then |Z| is positive normal.


    2. Relevant equations



    3. The attempt at a solution
    I am not really sure where to begin with this. Can anyone provide me a jumping off point, please?

    I do know that [tex]\phi[/tex](-x) = 1 - [tex]\phi[/tex](x)
    and so [tex]\phi[/tex](x) + [tex]\phi[/tex](-x) - 1 = 0.
    I am not sure how to utilize that or if it is even on the right track. Thanks for any help.
     
  2. jcsd
  3. Mar 30, 2009 #2

    statdad

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    Homework Helper

    Start with the cumulative distribution function of [tex] |Z| [/tex]

    [tex]
    P(|Z| \le x) = P(-x \le Z \le x) = \Phi(x) - \Phi(-x)
    [/tex]

    where [tex] \Phi [/tex] is the cdf of the standard normal. How can you simplify [tex] \Phi(-x) [/tex]?
     
  4. Mar 30, 2009 #3
    Ahhh, that makes perfect sense statdad. I kept trying to factor out 'x' when I looked at it the way you did it and got nowhere so discarded that method. I would substitute that with the equation I put in my first post to get the equation from the definition of positive normal. I just could not come up with that middle step to link the ideas until you put it so plainly. Thanks!


     
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