(See Solution) Let X be a continuous random variable with density f_X(x)=(1)/(2 σ) \exp [-(|x-μ|)/(σ)], -∞ First we will derive the

Question: Let $X$ be a continuous random variable with density

\[f_{X}(x)=\frac{1}{2 \sigma} \exp \left[-\frac{|x-\mu|}{\sigma}\right], \quad-\infty

First we will derive the moment generating function of $X, m_{X}(u)$.

Show that:
\[m_{X}(u)=\frac{e^{u \mu}}{1-u^{2} \sigma^{2}}\]
Find the range of values of $u$ for which $m_{X}(u)$ is the moment generating function.

(b) For a random sample of size $n$ from $X$,

Find the method-of-moment estimates of $\mu$ and $\sigma .$ Hint: use $m_{X}(u)$.
$^{*}$ Derive the maximum likelihood estimators for $\mu$ and $\sigma$.

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Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document

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(See Solution) Let X be a continuous random variable with density f_X(x)=(1)/(2 σ) \exp [-(|x-μ|)/(σ)], -∞ First we will derive the

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