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 \begin{center}
  {\Large{\bf {Problem Set-2}}}\\
  {Dated: 23/08/2018 \& Submission deadline is on: 30/8/2018}
   \end{center}


\begin{enumerate}

\item Let $X$ be a random variable having the p.m.f.
\begin{align*}
p(x) = \left\{\begin{array}{ll} 
                     \frac{c}{(2x-1)(2x+1)} & \text{if } x \in \{1, 2, 3, \cdots\}\\
                    0 & \text{otherwise}
            \end{array}\right.
\end{align*} where $c$ is a real constant.
\begin{enumerate}
\item Find the value of constant $c$.
\item Find the cumulative distribution function of $X$.
\item For the positive integers $m$ and $n$ such that $m < n$, evaluate $P(X < m+1), P(X \geq m), P(m \leq X < n)$ and $P(m < X \leq n)$.
\item Find the conditional probabilities $P (\{X > 1\}|\{1 \leq X < 4\})$ and $P (\{1 < X < 6\}|\{X \geq 3\})$.
\end{enumerate}
\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 The exponential distribution for random variable $X$ is given by 
      
      \begin{equation*}
       f(x) = \lambda e^{-\lambda x}, \qquad  x \geq 0
      \end{equation*}
Find the mean of $x$. Find the moment generating function $M_X(t)$ of the p.d.f.\\

%\item joint distribution problems

\item Suppose the random variable X has distribution function
\begin{align*}
F(x) = \left\{\begin{array}{ll} 
                   A e^{-x} & 0<x<\infty\\ 
                    0 & \text{otherwise}
                                \end{array}\right.
\end{align*}
Find the value of the constant A and hence calculate the probability that X lies
in the interval $1 < X < 2$.

\newpage
\item Roll a dice $(n=1,2,...6)$. Two events $s_1$ and $s_2$ are defined as follows:\\

\[s_1=
 \begin{cases}
1  & \text{if } n=2,4,6  \\
-1 & \text{if } n=1,3,5
\end{cases}
\]

\[s_2=
\begin{cases}
1  & \text{if } n=3, 6  \\
-1 & \text{if } n=1, 2, 4, 5
\end{cases}
\]

Show that $<s_1s_2>=<s_1><s_2>$. Show that $P(s_1,s_2)=P_1(s_1)P_2(s_2)$. So $s_1$ and $s_2$ are uncorrelated. 

\item Repeat $1$ with the following $s_1$ and $s_2$ to show that the events are correlated. Find $Cov(s_1,s_2)$ and correlation coefficient.

\[s_1=
 \begin{cases}
1  & \text{if } n=1, 2, 3  \\
-1 & \text{if } n=4, 5, 6
\end{cases}
\]

\[s_2=
\begin{cases}
1  & \text{if } n=2, 4, 6  \\
-1 & \text{if } n=1, 3, 5
\end{cases}
\]
\item (i) Suppose a group of $100$ men aged $60-64$ in Dehradun received a new flu vaccine from a health center in $2014$. From the $2014$ life table of the health center, it is found that the approximate probability that a man, aged between $60-64$, dies in the next year is $0.02$. How likely are, at least $5$ out of $100$ men who received flu vaccine and aged $60-64$ to die within the next year? \\
(ii) What is the probability that amongst the $60$ to $64$-year old men who got flu vaccination exactly $25$ survive and at least $10$ die within the next year? ({\it you don't need to calculate the exact numerical values of the probabilities})\\
\end{enumerate}

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