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\begin{document}
\title{Stat 150 Homework \#1}
\author{Spring 2022}
\maketitle

\textbf{Due Friday, February 4th, at 11:59 PM on Gradescope.}\\

Note that there are \emph{Exercises} and \emph{Problems} in the textbook. Make sure you read the homework carefully to find the assigned question.

\bigskip{}

\begin{itemize} \setlength{\itemsep}{10pt}

\item[1.] Pinsky and Karlin, Problem 1.3.11
  
\item[2.] Pinsky and Karlin, Problem 1.4.3

\item[3.] Pinsky and Karlin, Problem 1.5.7 

\item[4.] Pinsky and Karlin, Problem 1.5.8

\item[5.] Pinsky and Karlin, Problem 2.1.2

\item[6.] Pinsky and Karlin, Problem 2.1.9
 
\item[7.] Pinsky and Karlin, Problem 2.3.5
  
\item[8.] Let $X$ be a random variable. Recall that the moment generating function (or MGF for short) $M_X(t)$ of $X$ is the function $M_X: \R \to [0, \infty]$ defined by $t \mapsto \E[e^{tX}]$. Now suppose that $X \sim \text{Gamma}(\alpha, \lambda)$, where $\alpha, \lambda > 0$. Prove that
    \[
      M_X(t) = \begin{dcases}
        \left(\frac{\lambda}{\lambda - t}\right)^\alpha &\text{if } t < \lambda; \\
        \infty &\text{if } t \geq \lambda.
      \end{dcases}
    \]
\end{itemize}

\end{document}