% 256f19Assignment5.tex             Continuous distributions
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{\Large \textbf{\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19}{STA 256f19} 
Assignment Five}}\footnote{Copyright information is at the end of the last page.}
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\noindent
Please read Section 2.4 and the first part of 2.5 (pages 51-68) in the text. You can skip 2.5.4 and 2.5.5. Also, look over your lecture notes. The following homework problems are not to be handed in. They are preparation for Term Test 2 and the final exam.  Use the formula sheet.

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\begin{enumerate} 

\item Do Exercise 2.4.1 in the text. % cdf of U(0,1)

\item Let $X \sim$ Exponential($\lambda$). 
    \begin{enumerate}
        \item Find the cumulative distribution function $F_{_X}(x)$. Be sure it is defined for all real $x$.
        \item Do Exercise 2.4.3, parts a-c only. The answer to (d) in the book is incorrect
    \end{enumerate}

\item Do Exercise 2.4.5 in the text. % Neg can't be density.
\item Do Exercise 2.4.7 in the text. % Find c.
\item Do Exercise 2.4.9 in the text. % One density greater; integrate both sides.
\item Do Exercise 2.4.10 in the text. % One density greater; integrate both sides.
\item Do Exercise 2.4.14 through 2.4.19 in the text. % Includes gamma integrates to one.

\item Do Exercise 2.4.21 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. This is useful because it helps us recognize the limitations of our knowledge.

\item The continuous random variable $X$ has density 
$    f_{_X}(x) = \left\{ \begin{array}{ll}
   \frac{\alpha}{x^{\alpha+1}} & \mbox{for $ x \geq 1$}  \\
   0     & \mbox{for } x<1 
   \end{array}  \right. $,  % Need that crazy invisible right period!
where $\alpha > 0$. \\  Let $Y = 1/X$. Find the density of $Y$. Be careful to specify where it is non-zero.
If you look at it carefully, you will see that this is a beta distribution with $\beta=1$.

\item Let the continuous random variable $X$ have cumulative distribution function $F(x) = \left\{ \begin{array}{ll} % ll means left left
     0 & \mbox{for $ x < 0$}  \\
   x^3 & \mbox{for $ 0 \leq x \leq 1$}  \\
   1     & \mbox{for } x>1 
   \end{array}  \right. $. 
Find the density $f(x)$. Be careful to specify where it is non-zero.

\item The density of the normal distribution with parameters $\mu$ and $\sigma^2$ is given on the formula sheet. 
    \begin{enumerate}
        \item Show that $f(x)$ is symmetric about $\mu$, meaning $f(\mu+x)=f(\mu-x)$.
        \item Let $X\sim$ N($\mu,\sigma$) and $Z = \frac{X-\mu}{\sigma}$. Find the density of $Z$. Where is it non-zero?
        \item Let $X\sim$ N($\mu,\sigma$) and $Y = aX+b$, where $a$ and $b$ are constants, and $a \neq 0$. Find the density of $Y$. Where is it non-zero? Do you recognize this density?
        \item Let $Z \sim N(0,1)$. This is the standard normal, in which $\mu=0$ and $\sigma^2=1$.  If $x>0$, show $F_{_Z}(-x) = 1-F_{_Z}(x)$. Hint: Write $F_{_Z}(-x)$ as an integral and do a change of variables. 
        \item Let $X \sim N(\mu=50,\sigma^2 = 100)$. For the following, use Table D2 on page 712. It will be provided with the test and final exam. I suggest drawing pictures.
            \begin{enumerate}
                \item Find $P(X<60)$. [0.8413]
                \item Find $P(X>30)$. [0.9772]
                \item Find $P(30<X<55)$.  [0.6687]
            \end{enumerate}
        \item Let $Z \sim N(0,1)$, the standard normal. Show that the standard normal density integrates to one. Hint: Split the integral at zero and let $t = \frac{z^2}{2}$. You may use $\Gamma(\frac{1}{2}) = \sqrt{\pi}$ for now without proof.
        \item Using the last result, show that the general normal density integrates to one.
        \item Let $Z \sim N(0,1)$ and $Y=Z^2$.
            \begin{enumerate}
                \item For what values of $y$ is $f_y(y)>0$?
                \item Show that $Y$ has a gamma distribution and give the parameters. Again, you may use the fact that $\Gamma\left( \frac{1}{2}\right) = \sqrt{\pi}$, without proof.
            \end{enumerate}
    \end{enumerate} % End of normal questions.


\item Let $X$ have an Exponential($\lambda$) distribution, and let $Y=\lambda X$. Find the density of $Y$. Be sure to specify where it is non-zero.

\item Let the continuous random variable $X$ have distribution function $F_{_X}(x)$, and let $Y=F_{_X}(X)$. That's right. You are transforming a random variable by its own cumulative distribution function. 
    \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 continuous uniform random variable on the interval from zero to one. Find the cumulative distribution function and density of $Y$. 

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\noindent
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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