For the random variable X with the given probability mass function as below, find the mean and variance.
\(f(x)=\begin{cases} \begin{matrix} \cfrac { 1 }{ 2 } e^{ -\frac { x }{ 2 } } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)

If X~ B(n, p) such that 4P(X = 4) = P(X = 2) and n = 6. Find the distribution, mean and standard deviation of X.

An urn contains 2 white balls and 3 red balls. A sample of 3 balls are chosen at random from the urn. If X denotes the number of red balls chosen, find the values taken by the random variable X and its number of inverse images

Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.

Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.

A pair of fair dice is rolled once. Find the probability mass function to get the number of fours.

If the probability mass function f(x) of a random variable X is

x	1	2	3	4
f (x)	\(\cfrac { 1 }{ 12 } \)	\(\cfrac { 5 }{ 12 } \)	\(\cfrac { 5 }{ 12 } \)	\(\cfrac { 1 }{ 12 } \)

find (i) its cumulative distribution function, hence find
(ii) P(X ≤ 3) and,
(iii) P(X ≥ 2)

Find the probability mass function f(x) of the discrete random variable X whose cumulative distribution function F(x) is given by
 
Also find
(i) P(X < 0) and
(ii) P(\(X \geq-1)\)

If X is the random variable with distribution function F(x) given by,
\(F(x)=\begin{cases} \begin{matrix} 0 & x<0 \end{matrix} \\ \begin{matrix} x & 0\le x<1 \end{matrix} \\ \begin{matrix} 1 & 1\le x \end{matrix} \end{cases}\) 
then find
(i) the probability density function f(x)
(ii) P(0.2 ≤ X ≤ 0.7)

Find the mean and variance of a random variable X , whose probability density function is \(f(x)=\begin{cases} \begin{matrix} { \lambda e }^{ -2x } & for\ge 0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)