Let $$M(t)=\frac{5}{1-8t}$$ for $t<1/8$ be the mgf of random variable $X$. Find $E(X)$ and $Var(x)$.
I am not sure how to use the mgf to find the $E(X)$. Once I have the expected value I can find the variance.
Thank you in advance!
$\endgroup$ 12 Answers
$\begingroup$Hint: For any $n\in \mathbb N$ $$E\left(X^{n}\right)=M_{X}^{(n)}(0)={\frac {d^{n}M_{X}}{dt^{n}}}(0)$$
$\endgroup$ $\begingroup$Find $E(X)=\frac{d}{dt}M(t)|_{t=0}$ and $E(X^{2})=\frac{d^{2}}{dt^{2}}M(t)|_{t=0}$. Sustitute these values into the definition of $Var(X)$.
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