If X is N ( μ ; σ 2 ), then show that Z = ( X − μ ) σ is a standard normal random variable; that is, N ( 0 ; 1 )Here is how far I have gotten: P ( Z ≤ z ) = P ( X ≤ z σ + μ ) = ∫ x − μ σ z e − y 2 2 2 π d y .I used x = y σ + μ as my change of variable.1.) Is my result correct so far?2.) And what is my next step? This integral is out of my scope.

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If   X is   N  (  μ  ;      σ    2    ), then show that   Z  =            (      X      −      μ      )        σ   is a standard normal random variable; that is,   N  (  0  ;  1  )Here is how far I have gotten:  P  (  Z  ≤  z  )  =  P  (  X  ≤  z  σ  +  μ  )  =      ∫                            x          −          μ                σ                    z                    e                                    −                          y              2                                2                            2      π              d    y  .I used   x  =  y  σ  +  μ as my change of variable.1.) Is my result correct so far?2.) And what is my next step? This integral is out of my scope.

ambivalentnoe1 · Accepted Answer

The rightmost expression should be      ∫          −      ∞              z      σ      +      μ            1          σ              2        π              exp  −            (      x      −      μ              )        2                    2              σ        2                  d    x  .No substitution is requiblack to obtain this, just the definition of   N  (  μ  ,        σ    2    ). The exercise is to rewrite this as      ∫          −      ∞        z        1          2      π        exp  −            z      2        2        d    zwith the substitution   z  =            x      −      μ        σ  .

I'm struggling with this question: If X is N(mu;sigma^2), then show that Z=(X−mu)sigma is a standard normal random variable; that is, N(0;1)

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