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Let X have moment generating function M(t), and let v(t)=lnM(t). show that v'(0)=E(X) and v''(0)=Var(X).

I know the formula's for E(X) and Var(X), but don't I need an original pdf of X to compute the expected value and variance? How would I show these are equal without an original equation or without M(t)?

user141745
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  • Do you know formulas for the expected value and variance in terms of the moment generating function? – Dustan Levenstein Apr 29 '14 at 22:19
  • No, just their formulas when given a pdf – user141745 Apr 29 '14 at 22:24
  • well, I'm pretty sure that's what this problem is going for. Specifically, $E(X) = M'(0)$ and $E(X^2) = M''(0)$. – Dustan Levenstein Apr 29 '14 at 22:26
  • And the variance can be given in terms of $E(X)$ and $E(X^2)$. – Dustan Levenstein Apr 29 '14 at 22:27
  • right, what I'm asking is how I prove this when I am not given a pdf? Don't I need a pdf so I can do the calculations to show what E(X) is and Var(X) is, and then calculate through the mgf that M'(0) is the same as E(X) and M''(0) is the same as Var(X)? – user141745 Apr 29 '14 at 22:41
  • If you really feel the need to start from first principles, call your pdf $f$. Now you have a pdf you can do the MGF transform on. This is a good question, by the way. – nomen Apr 29 '14 at 22:45

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