Let X be random variable with probability density function
\(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
Which of the following statement is correct 

(a)

both mean and variance exist

(b)

mean exists but variance does not exist

(c)

both mean and variance do not exist

(d)

variance exists but Mean does not exist

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

(a)

6

(b)

4

(c)

3

(d)

2

If the function \(f(x)=\frac { 1 }{ 12 } \) for a < x < b, represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b?

(a)

0 and 12

(b)

5 and 17

(c)

7 and 19

(d)

16 and 24

Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E(X) is

(a)

0.11

(b)

1.1

(c)

11

(d)

1

If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).

(a)

\(\frac { 2 }{ 3 } \)

(b)

\(\frac { 2 }{ 5 } \)

(c)

\(\frac { 1 }{ 5 } \)

(d)

\(\frac { 1 }{ 3 } \)

The probability mass function of a random variable is defined as:

x	-2	-1	0	1	2
f(x)	k	2k	3k	4k	5k

Then E(X ) is equal to: 

(a)

\(\frac { 1 }{ 15 } \)

(b)

\(\frac { 1 }{ 10 } \)

(c)

\(\frac { 1 }{ 3 } \)

(d)

\(\frac { 2 }{ 3 } \)

If in 6 trials, X is a binomial variable which follows the relation 9P(X = 4) = P(X = 2), then the probability of success is

(a)

0.125

(b)

0.25

(c)

0.375

(d)

0.75

If F(x) is the probability distribution function then \(F\left( -\infty \right) \) is is _____________

(a)

1

(b)

2

(c)

\(\infty \)

(d)

0

If \(f(x)=\frac { 1 }{ 2 } \) ,\(E\left( { x }^{ 2 } \right) =\frac { 1 }{ 4 } \) then var(x) is _____________

(a)

0

(b)

\(\frac { 1 }{ 4 } \)

(c)

\(\frac { 1 }{ 2 } \)

(d)

1

If a random variable X has the p.d.f.\(f(x)=\frac { k }{ { x }^{ 2 }+1 }\) ,0 then k is _____________

(a)

\(\pi \)

(b)

\(\frac { 1 }{ \pi } \)

(c)

1

(d)

\(\frac { 2 }{ \pi } \)