Server : Apache System : Linux profile 3.10.0-1160.88.1.el7.x86_64 #1 SMP Tue Mar 7 15:41:52 UTC 2023 x86_64 User : apache ( 48) PHP Version : 8.0.28 Disable Function : NONE Directory :  /var/www/html/srijit/sbda131-18/

Upload File : current_dir [ Writeable ] document_root [ Writeable ]

\documentclass[12 pt]{article}%\makeatletter
\oddsidemargin .0in \evensidemargin .0in \textwidth 6.5in
\topmargin -.30in \textheight 24cm
\usepackage{amsmath,amssymb,amsthm,mathrsfs,booktabs,graphics,graphicx,float,mathtools,color}
\usepackage[mathscr]{euscript}
\usepackage{setspace}
%\usepackage{enumitem}
\newcommand{\boldr}{\boldsymbol{r}}
\newcommand{\boldi}{\boldsymbol{i}}
\newcommand{\boldj}{\boldsymbol{j}}
\newcommand{\boldk}{\boldsymbol{k}}
\newcommand{\boldv}{\boldsymbol{v}}
\newcommand{\rt}[1]{\textcolor{red}{#1}}
%\newcommand{\rt}[1]{\textcolor{red}{#1}}
\newcommand{\rh}[1]{\hfill{\rt{{#1}}}}

% =============================================================== %
%      List environment with lower case alphabetic counter        %
% =============================================================== %
\newcounter{list-counter}
\newenvironment{listalph}
{\begin{list}{(\alph{list-counter})~}{\usecounter{list-counter}}
%\setlength{\itemsep}{-0.1in}
%\setlength{\topsep}{0in}
}
{\end{list}}

\begin{document}

\parskip = 8pt

\begin{center}
\large{\textbf{Indian Institute of Information Technology Allahabad}}
\end{center}
\vspace{-24pt}
\begin{center}
\large{\textbf{Review Test (C1) on Biological Data Analytics}}
\end{center}

\hfill \footnotesize{Date: 19/09/2018}
%\vspace{0.05cm}
\vspace{-15pt}

\noindent
\hrulefill
\vspace{-16pt}

\begin{center}
\footnotesize{Paper Code: SBDA131C}
\end{center}
\vspace{-20pt}

%\begin{center}
%\footnotesize{Paper Setter: Abdullah Bin Abu Baker \& Sumit Kumar Upadhyay}
%\end{center}
\vspace{-10pt}
\noindent
\footnotesize{Max Marks: 20} \hfill \small{Duration: 75 Minutes}

\noindent
\footnotesize{Attempt \textbf{all} the questions. There is \textbf{no credit} for an answer if proper justification is not given, even if the answer is correct. Notations are standard. Calculator is {\bf not } allowed. Standard Normal table is given at the back side.}

\noindent
\hrule

\singlespacing
\thispagestyle{empty}
\begin{enumerate}

 \item Births in a Children's Hospital of a city occur randomly at an average rate of $2$ births per hour.    \begin{enumerate}
    \item Determine the probability of occurring $4$ births in a given hour of the hospital.
    \item What is the probability of observing more than or equal to $3$ births in a given
hour at the hospital?  
\item What is the probability that we observe $5$ births in a given $2$ hour interval?  
\item Determine the $3$rd central moment ($\mu_3=E[(X-\mu)^3]$) for the random variable chosen to solve the problem. Argue that ${\mu_3 \over \sigma^3}$ is a decreasing function of the mean $\mu$. $\sigma$ is the standard deviation of the random variable. 

\item Now suppose we know that in another Child hospital births occur randomly at an average rate
of $2.3$ births per hour. What is the probability that we observe $7$ births in total from the two hospitals in a given $1$ hour period?
    \item Given that
$5\%$
of the newborns in the city are $O^{+}$, estimate an approximate
 probability that a random sample of $100$ children contains $2$ or more
$O^{+}$ child.  \hfill[2+1+1+4+1+2=11]


\end{enumerate}
\item Suppose a particular disease is caused due to $4$ types (A, B, C, D) of bacterias of same group. In a given sample, probabilities of occurrence of each of these bacterias are equal. Every time in a particular sample one finds $A$, it is seen $B$ and $C$ types are also present- upon performing a confirmation test $T_1$. What is the probability of getting at least $3$, $D$ type bacterias in $5$ samples?  \\

Now using another confirmation test $T_2$, one always gets either $A-D$, or $B-C$ together in a given sample. Find out correlation coefficient between the random variables corresponding to the possible results/outcomes of two confirmation tests. \hfill[7]

\item The Uptimer is a custom-made lightweight battery-operated activity monitor that records the amount of time an individual spends in the upright position. In a study of children ages 8 to 15 years, Eldridge et al. studied 529 normally developing children who each wore the Uptimer continuously for a 24-hour period that included a typical school day. The researchers found that the amount of time children spent in the upright position followed a normal distribution with a mean of 5.73 hours and standard deviation of 1.3 hours. Assume that this finding applies to all children 8 to 15 years of age. Find the probability that a child selected at random spends less than 3 hours in the upright position in a 24-hour period. \hfill[2]

\end{enumerate}
\vspace{2.5cm}
\end{document}