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Math Help - random variable of continuous type

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    random variable of continuous type

    Let R(t)= ln M(t), where M(t) is the moment-generating function of a random variable of the continuous type. How do you show the following?
    a) \mu=R'(0)
    b) \sigma^{2}=R''(0)
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    Hello,
    Quote Originally Posted by antman View Post
    Let R(t)= ln M(t), where M(t) is the moment-generating function of a random variable of the continuous type. How do you show the following?
    a) \mu=R'(0)
    b) \sigma^{2}=R''(0)
    We know that M(0)=1~,~M'(0)=\mu~,~M''(0)=\mathbb{E}(X^2)
    And remember that \sigma^2=\mathbb{E}(X^2)-\mu^2

    a)
    R'(t)=\frac{M'(t)}{M(t)} \Rightarrow R'(0)=\frac{\mu}{1}=\mu

    b)
    R''(t)=\frac{M''(t)M(t)-M'(t)M'(t)}{M(t)^2} (using quotient rule for \frac{M'(t)}{M(t)})

    Thus R''(0)=\frac{\mathbb{E}(X^2) \cdot 1-\mu^2}{1^2}=\dots

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