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{\Large \textbf{STA 256f18 Assignment Five}}\footnote{Copyright information is at the end of the last page.}
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Please read Sections 2.2 and 2.3 in Chapter 2 of the textbook and look over your lecture notes. These homework problems are not to be handed in. They are preparation for Term Test 2 and the final exam.  All textbook problems are from Chapter Two. Use the formula sheet to do the problems. You will have a copy of Table~2 from Appendix~B in the text. On tests and the final exam, you may use anything on the formula sheet unless you are being directly asked to prove it.

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%%%%%%%%%%%%%%%%%%%% Continuous RVs in general %%%%%%%%%%%%%%%%%%%%

\item Do problem 33 in the text. The criteria for $F(x)$ being a cumulative distribution function is at the bottom of p.~36.
\item  Do problem 34 in the text. For $f(x)$ to be a density, it must be non-negative and integrate to one. The first quartile is that point $q_1$ such that $P(X \leq q_1) = \frac{1}{4}$. The third quartile is that point $q_3$ such that $P(X \leq q_3) = \frac{3}{4}$. The second quartile is the median.
\item For the distribution of problem 34 in the text, give
    \begin{enumerate}
        \item $F(-2)$ \hspace{5mm} [Answer: 0]
        \item $F(-1)$ \hspace{5mm} [Answer: 0]
        \item $F(0)$~~ \hspace{5mm} [Answer: $\frac{1}{2} - \frac{\alpha}{4}$]
        \item $F(1)$~~ \hspace{5mm} [Answer: 1]
        \item $F(2)$~~ \hspace{5mm} [Answer: 1]
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\item Do problem 38 in the text. 
\item Do problem 39 in the text. ``Recall" that $\frac{d}{dx} \tan^{-1}(x) = \frac{1}{1+x^2}$. The Cauchy distribution is the problem child of Statistics. Frequently, results that seem to be true in general are not true for the Cauchy. Thus it is useful because it helps us recognize the limitations of our knowledge.
\item Do problem 40 in the text.

%%%%%%%%%%%%%%%%%%%%  Common Continuous distributions %%%%%%%%%%%%%%%%%%%%

\item  Do problem 35 in the text. % uniform
\item Do problem 37 in the text. % uniform
\item Do problem 41 in the text.  % Exponential quartiles - drop?
\item Do problem 44 in the text. Also show that the probabilities sum to one.
\item Do problem 45 in the text. % Baby exponential.
\item If $X$ is an exponential random variable, prove $P(X>t+s|X>s) = P(X>t)$, where  $t>0$ and $s>0$. % From sample prolems
\item Do problem 48 in the text. % Gamma integrates to one. % Also in sample problems. 
\item Let $X\sim$ N($\mu,\sigma$) and $Z = \frac{X-\mu}{\sigma}$. Find the density of $Z$.
\item Do problem 52 in the text. There are 12 inches in a foot and 2.54 centimeters in an inch. Use Table~2 from Appendix~B in the text for part (a).
\item  Do problem 53 in the text. Use the table. % Normal with table.
\item  Do problem 54 in the text. Use the table. % Normal with table.
\item  Do problem 55 in the text. Use the table. % Normal with table.
\item  Do problem 56 in the text. % Normal 
\item  Do problem 57 in the text. % Normal 
\item  Do problem 61 in the text. % Gamma 
\item  Do problem 62 in the text. % General 

%%%%%%%%%%%%%%%%%%%%  Functions of a random variable. %%%%%%%%%%%%%%%%%%%%

\item  Do problem 62 in the text. % General 
\item  Do problem 68 in the text. % Transform exponential 
\item Let the continuous random variable $X$ have distribution function $F_x(x)$, and let $Y=F_x(X)$.
    \begin{enumerate}
        \item For what values of $y$ is $f_y(y)>0$? 
        \item Find $f_y(y)$. Do you recognize this distribution?
    \end{enumerate} 

\item Let the continuous random variable $X$ have cumulative distribution function $F_x(x)$ and density $f_x(x)$. The distribution function is strictly increasing on a single interval (which could be infinite), so that the inverse function $F_x^{-1}(y)$ is defined in the natural way. Let $Y = F_x^{-1}(U)$, where $U$ is a uniform random variable on the interval from zero to one. Find the cumulative distribution function and density of $Y$. 

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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
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\item Do Problem 8 in the text.

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