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Probability/Moment Generating Function

  1. Oct 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Let X ~ Normal(μ,σ2). Define Y=eX.
    a) Find the PDF of Y.
    b) Show that the moment generating function of Y doesn't exist.

    2. Relevant equations


    3. The attempt at a solution
    For part a, I used the fact that fy(y) = |d/dy g-1(y)| fx(g-1(y)). Therefore I got that fy(y)= (1/y)(1/√(2piσ2)e-(ln(y)-μ)2/2σ2

    Then for b), I used ψY(t)=E(etY)=∫etyfy(y)dy. When I plug in fy(y) I get a function that is nonlinear and too complicated to integrate. If someone could give me a hint on the next step it would be greatly appreciated.
     
  2. jcsd
  3. Oct 14, 2014 #2

    Ray Vickson

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    You need to analyze the behavior of ##e^{ty} f(y)## for ##y \to + \infty## in order to show that the integral does not converge. You do not need to actually compute the integral to do that.
     
  4. Oct 14, 2014 #3
    Where do I start in showing that (1/y)ety-ln(y)2/(2σ2)+2μln(y)/(2σ2) does not converge?
     
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