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\begin{document}
 \begin{center}
  {\Large{\bf {Problem Set-1}}}\\
   \end{center}

\rightline{Date: 05/08/2017}
\begin{enumerate}
\item Let E, F, G be three events. Find expressions for the events that of E, F, G
\begin{enumerate}
 \item only F occurs
 \item both E and F but not G occur
 \item at least one event occurs
 \item at least two events occur
 \item all three events occur
 \item none occurs
 \item at most one event occurs
 \item at most two events occur
\end{enumerate}

\item Let $S = \{0, 1, 2, · · · \}$ and $E ⊆ S$. Then in each of the following cases, verify P is a
probability on $S$.

\begin{enumerate}
 \item $P(E)=\sum_{x\in E}\frac{e^{-\lambda}\lambda^x}{x!}, \lambda >0$.
 \item $P(E)=\sum_{x\in E}p(1-p)^x, p>0$
 \item $P(E) = 0$, if $E$ has finite number of elements, and $P(E) = 1$, if $E$ has infinite
number of elements.
\end{enumerate}

\item Throw a die 20 times. What is the probability that you get a 6 on 10th trial?

\item Let E and F be two independent events. Then show that
\begin{enumerate}
 \item $E^c$ and F are independent.
 \item E and $F^c$ are independent.
 \item $E^c$ and $F^c$ are independent.
\end{enumerate}

\item Two digits are chosen at random without replacement from the set of integers ${1, 2, 3, 4, 5,
6, 7, 8}$. Find the probability that both digits are greater than 5. 

\item Consider two independent fair coins tosses, in which all four possible outcomes are
equally likely. Let H1 = {1st toss is a head}, H2 = {2nd toss is a head}, and D = {the
two tosses have different results}. Find $P(H1)$, $P(H2)$, $P(H1\cap H2)$, $P(H1|D)$, $P(H2|D)$,
and $P(H1\cap H2|D)$.

\item An urn contains b black balls and r red balls. One of the balls is drawn at random,
but when it is put back in the urn c additional balls of the same color are put in with
it. Now suppose that we draw another ball. Find the probability that the first ball
drawn was black given that the second ball drawn was red.

\item Suppose $84 \% $ of hypertensive and $23 \%$ of normotensive are classified as hypertensive by an automated blood-pressure machine. What are the predictive value positive and predictive value negative of the machine, assuming $20\%$ of the adult population is hypertensive?

\item The primary aim of a study by Carter et al. was to investigate the effect of the age at
onset of bipolar disorder on the course of the illness. One of the variables investigated was
family history of mood disorders. The table  shows the frequency of a family history of
mood disorders in the two groups of interest (Early age at onset defined to be 18 years or
younger and Later age at onset defined to be later than 18 years). Suppose we pick a person
at random from this sample. What is the probability that this person will be 18 years old
or younger?

\begin{tabular}{|c|c|c|c|}
 \hline\hline
Family History of Mood Disorders & Early=18(E) & Later $>$ 18(L) & Total \\
\hline
Negative (A) & 28 & 35 & 63\\
\hline
Bipolar Disorder (B) & 23 & 32 & 55\\
\hline
Unipolar (C) & 44 & 49 & 93\\
\hline
Unipolar and Bipolar (D) & 56 &62 & 118\\
\hline
Total & 151 & 178 & 339 \\

\hline\hline
\end{tabular}

Suppose we pick a subject at random from the $339$ subjects and find that he is $18$ years or
younger (E). What is the probability that this subject will be one who has no family history
of mood disorders (A)?

{\it Joint probability:} What is the probability that a person picked at random
from the $339$ subjects will be Early (E) and will be a person who has no family history of
mood disorders (A)?

 \item A bag contains ten balls. Among them six are red and four are white. Three balls are drawn at random and not replaced. Find the probability mass function for the number of red balls drawn.
 
%\item Biostat problems on Bayestheorem



%\item joint distribution problems

\item The quantity $0 \leq x \leq 1$ is distributed as $P(x)= Ax(1-x)$. What is A? What is the average value of $x$ and the standard deviation $s$? 

\item The joint density function of $X$ and $Y$ is given by
\begin{eqnarray*}
f (x, y) &=& 2e^{-x}  e^{-2y} ; \qquad 0 < x < \infty,~ 0 < y < \infty \\
         &=& 0 \qquad otherwise 
\end{eqnarray*}

Compute $(a) P\{X > 1, Y < 1\} ~; ~(b) P\{X < Y \}$.

\end{enumerate}

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