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

STA256 Fall 2018. 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 $p(12)$? The answer is a number. \vspace{10mm}
        \item What is $F(12)$? The answer is a number. \vspace{10mm}
        \item What is $p(27)$? The answer is a number. \vspace{10mm}
        \item What is $F(27)$? The answer is a number. \vspace{10mm}
        \item What is $p(4)$? The answer is a number. \vspace{10mm}
        \item What is $F(4)$? The answer is a number. \vspace{10mm}
        \item What is $F(4.5)$? The answer is a number. \vspace{10mm}
        \item What is $p(4.5)$? 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 $\Omega$, 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 $p(x)$ and $F(x)$ just for $x = 0, 1, 2$.  \vspace{50mm}
        \item What is $p(1.5)$? \vspace{8mm}
        \item What is $F(1.5)$? \vspace{8mm}
        \item What is $p(-9)$? \vspace{8mm}
        \item What is $F(-9)$? \vspace{8mm}
        \item What is $p(114)$? \vspace{8mm}
        \item What is $F(114)$? \pagebreak
        \item Graph $F(x)$.
    \end{enumerate}

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\item Let the discrete random variable $X$ have probability mass function $p(x) = cx$ 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 $p(0)$? The answer is a number.   \vspace{10mm}
        \item What is $F(0)$? The answer is a number.    \vspace{10mm}
        \item What is $F(5)$? The answer is a number.    \vspace{10mm}
        \item What is $p(2)$? The answer is a number.    \vspace{10mm}
        \item What is $F(1)$? 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 $p(0)$ 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 
\item 
    \begin{enumerate}
        \item Your relationship will last at least one year. The answer is a number. \vspace{100mm} 
        % 0.999^365 = 0.694
        \item Your relationship will last between one year and two years. That is if $X$ is the day on which the relationship ends, what is $P(366 \leq X \leq 730)$? % P(X geq 365) - P(X geq 731) = .694 - .482 = 0.212
    \end{enumerate}

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\item The boss says three strikes and you're out. If you are late to work 3 times, you're fired. If the probability of being late to work is $p$ and being late or not on each day are independent events, what is the probability of being fired on day $k$? The answer is a symbolic expression.

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\end{enumerate}

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\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/256f18} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f18}}
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