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- Suppose $X$ has moment generating function $M_X(t) = \frac{1}{(1 - 2t)^3}$ for $t < \frac{1}{2}$. Find the mean and variance of $X$. | - Suppose $X$ has moment generating function $M_X(t) = \frac{1}{(1 - 2t)^3}$ for $t < \frac{1}{2}$. Find the mean and variance of $X$. | ||
- Suppose $X \sim \text{Expo}(\lambda)$. Use the moment generating function of $X$ to show that the $k$th moment of $X$ is $\mu_k = \frac{k!}{\lambda^k}$. | - Suppose $X \sim \text{Expo}(\lambda)$. Use the moment generating function of $X$ to show that the $k$th moment of $X$ is $\mu_k = \frac{k!}{\lambda^k}$. | ||
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+ | **For Problem Set due 2 November**: 18 | ||
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+ | Answers: | ||
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+ | - $M_X(t) = \frac{1}{b - a}\int_a^b e^{tx}dx = \frac{e^{tb} - e^{ta}}{t(b - a)}$ | ||
+ | - $M_X(t) = \sum_{k=0}^\infty e^{tk}q^kp = p\sum_{k=0}^\infty (qe^t)^k = \frac{p}{1 - qe^t}$ provided $qe^t < 1$. | ||
+ | - $M_X' | ||
+ | - $M_X(t) = \frac{\lambda}{\lambda - t} = \frac{1}{1 - \frac{t}{\lambda}} = \sum_{k=0}^\infty \frac{t^k}{\lambda^k}$. Hence $\frac{\mu_k}{k!} = \frac{1}{\lambda^k}$. |