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\begin{document}
 \begin{center}
  {\Large{\bf {Problem Set-3}}}\\
 Special probability mass functions and density functions, covariance and correlation
   \end{center}

\rightline{Date:08/09/2017}
\begin{enumerate}
\item What is the probability of getting at least one $6$ in throwing of a dice $10$ times?
 
 \item  Toss a fair coin $M$ times. (a) Move one step forward (in one particular direction) each time you get a head (H). What is the probability $P(n)$ that you are n steps away from where you have started?
 
 (b) If you move one step forward for each H and one step backward for each tail (T), what is $P(n)$? What is the mean and variance of the probability mass function (PMF)? Find the mode of PF for $M=3$?
 
 \item Let $X$ be a Binomial random variable with parameters $n$, and $p$. Show that 

\[ P(X=x+1)=\frac{p}{1-p} \left(\frac{n-x}{x+1}\right)P(X=x)\].
 
 \item In a 10-over cricket match, the runs that can be scored by a poor batsman is
given by a Poisson distribution with parameter $\lambda = 10$. On
the other hand, the runs that a good batsman can score is given by a Poisson
distribution with parameter $\lambda = 30$. If a batsman scores 20 runs in the match,
would you judge him as good or poor?

\item Let $X$ be a Poisson random variable with parameter $\lambda > 0$. then show that $E(2^X) = \frac{1}{P(X = 0)}$
 
 \item An investigator notices that children develop chronic bronchitis in the first year of life in about 3 out of 20 households where both parents are chronic bronchitis, as compared to the national incidence rate of chronic bronchitis, which is $5\%$ in the first year of life. How likely are infants in at least 3 out of 20 households will develop chronic bronchitis 
 if probability of developing the disease in any one household is .05?
 
 \item A probability class has $300$ students
and each student has probability $1/3$ of getting an A, independently of any other
student. What is the mean of $X$, the number of students that get an A?

\item If $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$ , and if $a, b$
are scalars, then show that the random variable
\[Y=aX+b\] 

is also normal with mean $a\mu+b$ and variance $a^2\sigma^2$.

\item What is the probability that a $z$ picked at random from the population of $z$'s will have a
value between $-2.5$ and $2.5$?
 
 
 \item Two continuous random variables $X$ and $Y$ have a joint probability distribution function 
 \be
 f(x,y)=A(x+y),
 \ee
 where $A$ is a constant and $0\leq x \leq 1; 0\leq y \leq 1$. \\(a) Determine $A$.\\
 (b) Calculate the correlation (Cov$(X,Y)$) between $X$ and $Y$.
 
\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.

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

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

\begin{figure}[h]
 \centering
    \includegraphics[width=1\textwidth]{SNcdf}
    
      %\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}).}
   % \label{figure1}
\end{figure}
 
\end{enumerate}
\end{document}