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Normal random variables

  • Thread starter Proggy99
  • Start date
1. Homework Statement
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. Homework 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.
 

statdad

Homework Helper
1,489
30
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]?
 
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!


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]?
 

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