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\begin{center}   
{\Large \textbf{Sample Questions: Continuous Random Variables}}

STA256 Fall 2019. Copyright information is at the end of the last page.
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\begin{enumerate} 

\item The continuous random variable $X$ has density {\Large
$    f(x) = \left\{ \begin{array}{ll} % ll means left left
   \frac{c}{x^{\alpha+1}} & \mbox{for $ x \geq 1$}  \\
   0     & \mbox{for } x<1 
   \end{array}  \right. $  % Need that crazy invisible right period!
} % End size
where $\alpha > 0$.

    \begin{enumerate}
        \item Find the constant $c$ \vspace{30mm}
        \item Find the cumulative distribution function $F(x)$. \vspace{100mm}
        \item The median of this distribution is that point $m$ for which $P(X \leq m) = \frac{1}{2}$. What is the median? The answer is a function of $\alpha$.
    \end{enumerate} 

\pagebreak

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\item Let {\Large $F(x) = \left\{ \begin{array}{ll} % ll means left left
     0 & \mbox{for $ x < 0$}  \\
   x^\theta & \mbox{for $ 0 \leq x \leq 1$}  \\
   1     & \mbox{for } x>1 
   \end{array}  \right. $  % Need that crazy invisible right period!
} % End size

    \begin{enumerate}
        \item If $\theta=3$, what is $P\left(\frac{1}{2} < X \leq 4\right)$? The answer is a number. \vspace{30mm} % 7/8
        \item Find $f(x)$. 
    \end{enumerate} 

\pagebreak

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\item The Uniform$(L,R)$ distribution has density
{\large
    $f(x) = \left\{ \begin{array}{ll} % ll means left left
   \frac{1}{R-L} & \mbox{for $L \leq x \leq R$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.
   $  % Need that crazy invisible right period! $
} % End size
    \begin{enumerate}
        \item Give the cumulative distribution function. \vspace{120mm}
        \item Graph the cumulative distribution function.
    \end{enumerate}

\pagebreak

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\item The Exponential($\lambda$) distribution has density
{\Large
    $f(x) = \left\{ \begin{array}{ll} % ll means left left
   \lambda e^{-\lambda x}   & \mbox{for $x \geq 0$}  \\
   0     & \mbox{for } x < 0 
   \end{array}  \right.$  % Need that crazy invisible right period!
} % End size

    \begin{enumerate}
        \item Show $\int_{-\infty}^\infty f(x) \, dx = 1$  \vspace{60mm}
        \item Find $F(x)$ \pagebreak
    \end{enumerate} 

\pagebreak

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\item The Gamma($\alpha,\lambda$) distribution has density 
{\Large
    $f(x) = \left\{ \begin{array}{ll} % ll means left left
   \frac{\lambda^\alpha}{\Gamma(\alpha)} e^{-\lambda x} \, x^{\alpha-1}  & \mbox{for $x \geq 0$}  \\
   0     & \mbox{for } x < 0 
   \end{array}  \right.$  % Need that crazy invisible right period!
} % End size
\\ Show $\int_{-\infty}^\infty f(x) \, dx = 1$.

\pagebreak

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\item The Normal($\mu,\sigma$) distribution has density
{\Large
$f(x) = \frac{1}{\sigma \sqrt{2\pi}}\exp\left\{{-\frac{(x-\mu)^2}{2\sigma^2}}\right\}$
} % End size
    \begin{enumerate}
        \item Show that $f(x)$ is symmetric about $\mu$, meaning $f(\mu+x)=f(\mu-x)$. \vspace{70mm}
        \item Let $X\sim$ N($\mu,\sigma$) and $Z = \frac{X-\mu}{\sigma}$. Find the density of $Z$.
    \end{enumerate}

\pagebreak

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\item Let $Z \sim N(0,1)$ (standard normal), so that 
{\Large
$f_x(z) = \frac{1}{ \sqrt{2\pi}} e^{-\frac{z^2}{2}}$. If $x>0$, show $F_z(-x) = 1-F_z(x)$.
} % End size

\pagebreak

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\item Let $X \sim N(\mu=50,\sigma = 10)$. 
    \begin{enumerate}
        \item Find $P(X<60)$. The answer is a number. \vspace{70mm}
        \item Find $P(X>30)$. The answer is a number.  \vspace{70mm}
        \item Find $P(30<X<55)$.  % 0.6687
    \end{enumerate}
\pagebreak

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\item Let $Z \sim N(0,1)$ (standard normal), so that 
{\Large
$f_{_Z}(z) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{z^2}{2}}$. Show 
$\displaystyle \int_{-\infty}^\infty f_{_Z}(z) = 1$.
Hint: Let $t = \frac{z^2}{2}$. You may use $\Gamma(\frac{1}{2}) = \sqrt{\pi}$.
} % End size
% Let t = z^2/2

\pagebreak

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\item The beta density with parameters $\alpha$ and $\beta$ is 
%{\Large
    $f(x) = \left\{ \begin{array}{ll} % ll means left left
   \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \,
    x^{\alpha-1} (1-x)^{\beta-1} & \mbox{for $0 \le x \le 1$}  \\
   0     & \mbox{Otherwise} 
   \end{array}  \right.$  % Need that crazy invisible right period!
%} % End size
Let $X\sim$ Beta($\alpha,\beta$) with $\beta=1$.
    \begin{enumerate}
        \item Write the density of $X$ for $0 \leq x \leq 1.$  Simplify. You will prove $\Gamma(\alpha+1) = \alpha\Gamma(\alpha)$ in homework. 
        \vspace{30mm}
        \item  For what values of $y$ is $f_y(y)>0$? Show your work. \vspace{30mm}
        \item Derive $f_y(y)$. Don't forget to specify where the density is greater than zero.
    \end{enumerate} 

\pagebreak

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\item Let $Z \sim N(0,1)$ and $Y=Z^2$.
    \begin{enumerate}
        \item For what values of $y$ is $f_y(y)>0$? \vspace{20mm}
        \item Show that $Y$ has a gamma distribution and give the parameters. You may use the fact that $\Gamma\left( \frac{1}{2}\right) = \sqrt{\pi}$, without proof.
    \end{enumerate} 

\pagebreak

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\item In this problem, the random variable $X$ is transformed by its own distribution function. Let the continuous random vabriale $X$ have distribution function $F_x(x)$, and let $Y=F_x(X)$.
    \begin{enumerate}
        \item For what values of $y$ is $f_y(y)>0$? Hint: as $x$ ranges from $-\infty$ to $\infty$, $F_x(x)$ ranges from \underline{\hspace{8mm}} to  \underline{\hspace{8mm}}.   \vspace{2mm}
        \item Find $f_y(y)$. 
    \end{enumerate} 

\end{enumerate}

\vspace{140mm}

\noindent
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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/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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% The answer is a number. Circle your answer.

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\item If a random variable has density 
{\Large $f(x) = \frac{1}{2}e^{-|x|}$}, % End size
find the cumulative distribution function.

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        \item Still for the exponential density with $F(x) = 1-e^{-\lambda x}$ for $x \geq 0$, prove the ``memoryless" property:
\begin{displaymath}
    P(X>t+s|X>s) = P(X>t)
\end{displaymath}
for $t>0$ and $s>0$. For example, the probability that the conversation lasts at least $t$ \underline{more} minutes is the same as the probability of it lasting at least $t$ minutes in the first place.










