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{\Large \textbf{Sample Questions: Discrete Random Variables}}

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

\item Roll two fair dice. Let $X$ denote the sum of the two numbers.\vspace{60mm}
    \begin{enumerate}
        \item What is $p_{_X}(12)$? The answer is a number. \vspace{10mm}
        \item What is $F_{_X}(12)$? The answer is a number. \vspace{10mm}
        \item What is $p_{_X}(27)$? The answer is a number. \vspace{10mm}
        \item What is $F_{_X}(27)$? The answer is a number. \vspace{10mm}
        \item What is $p_{_X}(4)$? The answer is a number. \vspace{10mm}
        \item What is $F_{_X}(4)$? The answer is a number. \vspace{10mm}
        \item What is $F_{_X}(4.5)$? The answer is a number. \vspace{10mm}
        \item What is $p_{_X}(4.5)$? The answer is a number. 
    \end{enumerate} 

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\item A biased coin has $P(\mbox{Head}) = \frac{1}{3}$. Toss it twice.  
    \begin{enumerate}
        \item List the elements of the sample space $S$, together with their probabilities. \vspace{30mm}
        \item Let $X$ equal the number of heads. For what values of $x$ is $P(X=x)>0$?  \vspace{30mm}
        \item Give $p_{_X}(x)$ and $F_{_X}(x)$ just for $x = 0, 1, 2$.  \vspace{50mm}
        \item What is $p_{_X}(1.5)$? \vspace{8mm}
        \item What is $F_{_X}(1.5)$? \vspace{8mm}
        \item What is $p_{_X}(-9)$? \vspace{8mm}
        \item What is $F_{_X}(-9)$? \vspace{8mm}
        \item What is $p_{_X}(114)$? \vspace{8mm}
        \item What is $F_{_X}(114)$? \pagebreak
        \item {\Large Graph $F_{_X}(x)$. }
    \end{enumerate}

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\item For a general random variable $X$, {\Large prove $\displaystyle \lim_{x \rightarrow \infty} F_{_X}(x) = 1$. }

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\item For a general random variable $X$, {\Large prove $\displaystyle \lim_{x \rightarrow -\infty} F_{_X}(x) = 0$. }

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\item Let the discrete random variable $X$ have probability mass function $p_{_X}(x) = cx$ for $x=1,2,3,4$ and zero otherwise. What is the constant $c$?  \vspace{30mm}

\item Prove that the binomial probabilities sum to one. The formula sheet for Test 2 will have the Binomial Theorem: $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k}$  \vspace{50mm}


\item Let $X$ have a binomial distribution with $n=5$ and $p=\frac{1}{4}$. 
    \begin{enumerate}
        \item What is $p_{_X}(0)$? The answer is a number.   \vspace{10mm}
        \item What is $F_{_X}(0)$? The answer is a number.    \vspace{10mm}
        \item What is $F_{_X}(5)$? The answer is a number.    \vspace{10mm}
        \item What is $p_{_X}(2)$? The answer is a number.    \vspace{10mm}
        \item What is $F_{_X}(1)$? The answer is a number.    \vspace{10mm}
    \end{enumerate}
    
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\item \label{bumbrella} Cheap umbrellas are shipped to the dollar store in boxes of 20. The probability that the umbrella is defective (you can't even use it once) is 0.10. We will assume that being defective or not for the 20 umbrellas in a box are independent events, though this assumption may not be safe in practice, depending on the manufacturing and shipping process. 
    \begin{enumerate}
        \item What is the probability that all 20 umbrellas are okay? The answer is a number.  \vspace{30mm}
        \item Obtain that last number as $p_{_X}(0)$ for one of the standard probability distributions. \vspace{40mm}
        \item That is the probability that exactly two umbrellas are defective? The answer is a number.  \vspace{30mm}
        \item What is the probability that two or fewer umbrellas are defective? The answer is a number.  \vspace{30mm}
    \end{enumerate}

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\item In a box of 20 umbrellas, 2 are defective. If you sample 5 umbrellas randomly without replacement, what is the probability of at least one defective? The answer is a number. \vspace{30mm} 

\item Going back to the assumptions of Question~\ref{bumbrella}, the probability of a defective umbrella is 0.10, they are independent, and they are shipped in boxes 
of 20. You choose a box at random, and then sample 5 umbrellas randomly without replacement, what is the probability of at least one defective? The answer is a number.


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\item It is true love, but still the chances your significant other will break up with you on any given day is a tenth of one percent. Assuming 365 days in a year and independence, what is the probability that your relationship will last at least one year? The answer is a number.
        % 0.999^365 = 0.694

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\item Let $X_n$ have a binomial distribution with parameters $n$ and $\theta_n$. That is, 
$p_{_{X_n}}(x) = \binom{n}{x} \theta_n^x(1-\theta_n)^{n-x}$ for $x = 0, \ldots, n$. The probability $\theta_n$ depends on $n$. As $n \rightarrow \infty$, $\theta_n \rightarrow 0$  in such a way that $n\theta_n = \lambda$ remains constant. Find $\lim_{n \rightarrow\infty} p_{_{X_n}}(x)$, and identify the resultng distribution by name.

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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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