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 \begin{center}
  {\Large{\bf {Problem Set-2}}}\\
  joint probability, random variables, cumulative distribution function, (probability) density function/mass function, expectation, variance
   \end{center}

\rightline{Date:16/08/2017}
\begin{enumerate}
 \item Let $X$ be a random variable with probability mass function (pmf) $p(1)=.5, p(2)=.2$ and $ p(3)=.3$. Find the cdf.

 \item Are the following functions cumulative distribution (cdf) function?
\begin{enumerate}
\item $F(x) = \left\{\begin{array}{ll} 
                    0 & \text{if }x < 0\\ 
                    1- e^{-x} & \text{if }x \geq 0
            \end{array}\right.$
\item $F(x) = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}x, -\infty < x < \infty$.
\item $F(x) = \left\{\begin{array}{ll} 
                    0 & \text{if }x < -5\\ 
                    x & \text{if }-5 \leq x \leq 0.5\\
                    1 & \text{if } x > 0.5
            \end{array}\right.$
\end{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 cdf of $X$.
\end{enumerate}
\item Let $X$ be a random variable with p.m.f 
\begin{align*}
P(X = r) = {\displaystyle {n \choose r}} p^r (1-p)^{n-r},~ r = 0, 1, 2, \cdots, n,~ 0\leq p\leq 1.
\end{align*}
Find the p.m.f of the random variables (a) $Y = aX + b$ and (b) $Y = X^2$.
\item Let $X$ be a random variable denoting the outcomes of rolling of a die. Find the expectation and variance of random variable $Y=X^2$.

\item Suppose that 3 batteries are randomly chosen from a group of 3 new, 4
used but still working, and 5 defective batteries. If we let $X$ and $Y$ denote, respectively,
the number of new and used but still working batteries that are chosen, then find the joint
probability mass function of $X$ and $Y$.

\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 In a study by Cross et al., patients who were involved in problem gambling treatment were
asked about co-occurring drug and alcohol addictions. Let the discrete random variable $X$ represent
the number of co-occurring addictive substances used by the subjects. The table below  summarizes the
frequency distribution for this random variable.

\begin{tabular}{|c|c|}
 \hline\hline
Number of Substances Used & Frequency\\
\hline
0 & 144\\
1 & 342\\
2 & 142\\
3 & 72\\
4 & 39\\
5 & 20\\
6 & 6\\
7 & 9\\
8 & 2\\
9 & 1\\
\hline\hline
\end{tabular}
\begin{enumerate}
 \item Construct a table of the relative frequency and the cumulative frequency for this discrete
distribution.
\item What is probability that an individual selected at random used five addictive substances?
\item What is the probability that an individual selected at random used more than six addictive
substances?
\item What is the probability that an individual selected at random used between two and five addictive
substances, inclusive?
\item Find the mean, variance, and standard deviation of this frequency distribution.
\end{enumerate}

\item An exponential random variable $X$ has a probability density function of the form
\begin{align*}
f(x) = \left\{\begin{array}{ll} 
                     \lambda e^{-\lambda x} & if x\geq 0\\
                    0 & \text{otherwise}
            \end{array}\right.
\end{align*}
Find the variance of $X$. 
\end{enumerate}


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