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{\Large \textbf{Sample Questions: Transformations}}

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

    \item Let $X \sim$ Poisson($\lambda_1$) and $Y \sim$ Poisson($\lambda_2$) be independent. Using the convolution formula, find the probability mass function of $Z=X+Y$ and identify it by name.
    
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    \item Independently for $i = 1, \ldots, n$, let $X_i \sim$ Poisson($\lambda_i$), and let $\displaystyle Y_n = \sum_{i=1}^nX_i$. Using the last problem, what is the probability distribution of $Y_n$?
    
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    \item Let $X \sim$ Binomial($n_1,\theta$) and $Y \sim$ Binomial($n_2,\theta$) be independent. Using the convolution formula, find the probability mass function of $Z=X+Y$ and identify it by name.

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    \item Let $X_1, \ldots, X_n$ be independent Bernoulli random variables with parameter $\theta$, and let $\displaystyle Y_n = \sum_{i=1}^nX_i$. Using the last problem, what is the probability distribution of $Y$?
    
    

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    \item Let $X$ and $Y$ be independent exponential random variables with parameter $\lambda$. Using the convolution formula, find the probability density function of $Z=X+Y$ and identify it by name. 

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    \item Let $X_1$ and $X_2$ be independent standard normal random variables. Find the probability density function of $Y_1 = X_1/X_2$.


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    \item Use the Jacobian method to prove the convolution formula for continuous random variables.

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    \item Show that the normal probability density function integrates to one.

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    \item Prove $\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}$.  


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\item \label{Jacobian} The
random variables $X_1$ and $X_2$ are independent. $X_1$ has a gamma distribution with parameters $\alpha=a$ and $\lambda=1$, and $X_2$ has a gamma distribution with parameters $\alpha=b$ and $\lambda=1$. Let $Y_1=\frac{X_1}{X_1+X_2}$ and $Y_2=X_1+X_2$. 
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%        \item Solve for $x_1$ and $x_2$.
%        \item Calculate the determinant.
        \item \label{joint} Give the joint density of $Y_1$ and $Y_2$. Factor, separating $y_1$ and $y_2$ as much as possible. In your final statement of the answer to this part, \emph{specify where the joint density is non-zero.}

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        \item Find $f_{_{Y_1}}(y_1)$, the marginal density of $Y_1$. Again, do not forget to specify where the density is non-zero. \vspace{130mm}
        \item Identify the distribution of $Y_1$ by name; it is on the formula sheet. \vspace{10mm}
      %  \item Are $Y_1$ and $Y_2$ independent? Answer Yes or No and justify your answer in one sentence or less. 
    \end{enumerate}


\end{enumerate}

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This handout 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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    \item Continuing with
    {\Large
    $f_{x,y}(x,y) = \left\{ \begin{array}{ll} 
   2e^{-(x+y)} & \mbox{for $ 0 \leq x \leq y$ and $y \geq 0$}  \\
   0     & \mbox{otherwise}  
   \end{array}  \right. $
   
\noindent
Obtain $F_{xy}(x,y)$. Consider $x<y$ and $x>y$ separately.











