Let X be a random variable denoting the life time of an electrical equipment having probability density function
\(f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}\) 
Find
(i) the value of k
(ii) Distribution function 
(iii) P(X < 2)
(iv) calculate the probability that X is at least for four unit of time 
(v) P(X = 3)

Suppose that f (x) given below represents a probability mass function

x	1	2	3	4	5	6
f(x)	c2	2c2	3c2	4c2	c	2c

Find
(i) the value of c
(ii) Mean and variance.

Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs. 20 for each black ball selected and we lose Rs. 10 for each white ball selected. Find the expected winning amount and variance 

The mean and variance of a binomial variate X are respectively 2 and 1.5. Find 
(i) P(X = 0)
(ii) P(X =1)
(iii) P(X ≥1)

On the average, 20% of the products manufactured by ABC Company are found to be defective. If we select 6 of these products at random and X denote the number of defective products find the probability that
(i) two products are defective
(ii) at most one product is defective
(iii) at least two products are defective.