A random variable X has the density function $${\text{f}}\left( {\text{x}} \right) = {\text{K}}\frac{1}{{1 + {{\text{x}}^2}}},$$    where $$ - \infty $$  < x < $$\infty $$ . The value of K is

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that she studies mathematics the next day is 0.6. If she studies mathematics on a day, then the, probability that she studies computer science the next day is 0.4. Given that Aishwarya studies computer science on Monday, what is the probability that she studies computer science on Wednesday?

A. 0.24

B. 0.36

C. 0.4

D. 0.6

A coin is tossed 4 times. What is the probability of getting heads exactly 3 times?

A. $$\frac{1}{4}$$

B. $$\frac{3}{8}$$

C. $$\frac{1}{2}$$

D. $$\frac{3}{4}$$

A machine produces 0, 1 or 2 defective pieces in a day with associated probability of $$\frac{1}{6},\,\frac{2}{3}$$  and $$\frac{1}{6}$$ respectively. The mean value and the variance of the number of defective pieces produced by the machine in a day, respectively, are

A. 1 and $$\frac{1}{3}$$

B. $$\frac{1}{3}$$ and 1

C. 1 and $$\frac{4}{3}$$

D. $$\frac{1}{3}$$ and $$\frac{4}{3}$$

Let X and Y be two independent random variables. Which one of the relations between expectation (E), variance (Var) and covariance (Cov) given below is FALSE?

A. E(XY) = E(X) E(Y)

B. Cov (X, Y) = 0

C. Var (X + Y) = Var (X) + Var (Y)

D. E(X²Y²) = (E(X))² (E(Y))²