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  {\bf {\underline{INDIAN INSTITUTE OF INFORMATION TECHNOLOGY, ALLAHABAD}}}\\
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  {\small{\bf Mid-Semester Examination, September 2017}}
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\rightline{{\bf{\small Date of Examination: 24/ 09/17 (2nd Session)}} }
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{\small{\bf Program Code \& Semester: M.Tech.(BI), M. Tech.-PhD(BI)-1st Sem/ All PhD -PhD}}\\
{\small
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 {\bf Paper Title: Biological data analytics\\
  Paper Code: SBDA131C\\
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 Paper Setter: Dr. Srijit Bhattacharjee}
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{\bf \leftline{Max Marks: 30 \hspace{10cm} Duration: 2 hours}}

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Note: Use of non-programmable calculator is allowed. Be sure to carefully justify your answers. Total 5 problems are there each carrying 6 marks. The table of standard normal CDF values is provided. 
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\begin{enumerate}
 \item \begin{enumerate}
        \item In a certain high school class, consisting of $60$ girls and $40$ boys, it is observed that $24$ girls
and $16$ boys wear eyeglasses. If a student is picked at random from this class, the
probability that the student wears eyeglasses, $P(E)$, is $40/100$, or $.4$.
What is the probability that a student picked at random wears eyeglasses, given that
the student is a boy? Comment on the independence of the events being a boy and wearing eyeglasses. \hfill[2]
       
 \item Suppose $85 \% $ of hypertensive and $22 \%$ 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?\hfill[4]
  \end{enumerate}
  
  \item A biased die has probabilities $p/2,p,p,p,p,2p$ of showing $1,2,3,4,5,6$ respectively. Find $p$. Find the mean and variance of the outcomes. Compute the moment generating function for this probability mass function.
  
    \hfill[6]
  
  \item Roll a dice $(n=1,2,...6)$. Two events $s_1$ and $s_2$ are defined as follows:\\
  
  \[s_1=
 \begin{cases}
1  & \text{if } n=1, 2, 3  \\
-1 & \text{if } n=4, 5, 6
\end{cases}
\]

\[s_2=
\begin{cases}
1  & \text{if } n=2, 4, 6  \\
-1 & \text{if } n=1, 3, 5
\end{cases}
\]
Find $Cov(s_1,s_2)$ and correlation coefficient between $s_1$ and $s_2$. \hfill[6]

\item Diskin et al. studied common breath metabolites such as ammonia, acetone,
isoprene, ethanol, and acetaldehyde in five subjects over a period of 30 days. Each day,
breath samples were taken and analyzed in the early morning on arrival at the laboratory.
For subject A, a 27-year-old female, the ammonia concentration in parts per billion (ppb)
followed a normal distribution over 30 days with mean 491 and standard deviation 119.
What is the probability that on a random day, the subject’s ammonia concentration is
between 292 and 649 ppb? What is the probability that the subject's amonia concentration exceeds 391 ppb? \hfill[6]

\item The ideal size of a first-year class at a particular college is 150 students.
The college, knowing from past experience that, on the average, only 30 percent of those
accepted for admission will actually attend, uses a policy of approving the applications of
450 students. Compute the probability (approximate) that more than 150 first-year students attend this
college. Also compute the approximate probability that number of students attend the college lies between 120 and 280. \hfill[6]
\end{enumerate}
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\begin{figure}[t]
 \centering
    \includegraphics[width=1\textwidth]{standardnormaltable}
    
      %\caption{Penrose diagram of the spacetime formed by matching two charged Vaidya spacetimes along the thin null shell $S$ (thick green line). The thick red lines are the inner and the outer apparent horizons. The dotted lines are the event horizons of the background geometry (mass $m_f$, cf.~\eqref{mass_function}).}
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\leftline{Faculty Signature:}
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