\documentclass{amsart}

% PACKAGES

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb,enumerate}
\usepackage{amsthm,stmaryrd}
\usepackage[all]{xy}
\usepackage{hyperref}
\usepackage{multicol}
\usepackage{geometry}
\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}

\theoremstyle{definition}
\newtheorem{exer}{Exercise}

\newcommand{\bbr}{\mathbb{R}}
\newcommand{\bbc}{\mathbb{C}}
\newcommand{\bbz}{\mathbb{Z}}
\newcommand{\bbq}{\mathbb{Q}}
\newcommand{\bbn}{\mathbb{N}}

\begin{document}

\noindent
Statistics \\
MATH 371\\
Homework 0\\
Part 2 of 2.

\begin{exer}
Suppose that a point $(X,Y)$ is chosen from the disk 
\[
	S = \{(x,y)\,:\, (x-1)^2 + (y+2)^2\leq 9\}.
\]
Determine
\begin{enumerate}[(a)]
\item the conditional pdf of $Y$ for every given value of $X$.
\item $\operatorname{Pr}(Y>0 | X=2)$.
\end{enumerate}
\end{exer}

\begin{exer}
	Suppose that the pdf of $X$ is
	\[
		f(x) = 
			\begin{cases}
				\frac{1}{2}x & \text{for } 0<x<2,\\
				0 & \text{otherwise}.
			\end{cases}
	\]
	Determine the pdf of $Y = 4-X^2$.
\end{exer}

\begin{exer}
	Suppose that one word is to be selected at random from the sentence ``THE GIRL PUT ON HER BEAUTIFUL RED HAT.'' If $X$ denotes the number of letters in the word that is selected, what is the value of $E(X)$?
\end{exer}

\begin{exer}
	 Suppose that $X$ and $Y$ have a continuous joint distribution for which the jpdf is
	 \[
	 	f(x,y) = 
		\begin{cases}
			12y^2 & \text{for } 0\leq y\leq x\leq 1\\
			0 & \text{otherwise.}
		\end{cases}
	 \]
	 Find $E(XY)$.
\end{exer}

\begin{exer}
	Suppose that three random variables $X_1,X_2,X_3$ form a random sample from a distribution for which the mean is $5$. Determine the value of
	\[
		E(2X_1-3X_2+X_3-4).
	\]
\end{exer}

\begin{exer}
	Let $X$ be a random variable for which $E(X) = \mu$ and $\operatorname{Var}(X) = \sigma^2$. Show that
	\[
		E\left[ X(X-1) \right] = \mu(\mu -1)+\sigma^2.
	\]
\end{exer}

\begin{exer}
	Suppose that $X$ and $Y$ are independent random variables whose variances exist and such that $E(X) = E(Y)$. Show that
	\[
		E\left[ (X-Y)^2 \right] = \operatorname{Var}(X) + \operatorname{Var}(Y).
	\]
\end{exer}

\end{document}


\begin{exer}
\end{exer}















