If \(f(x)=\left\{\begin{array}{lc} k x^{2} & 0  if a p.d.f. then the value of k is ___________

(a)

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

(b)

\(\frac{1}{6}\)

(c)

\(\frac{1}{9}\)

(d)

\(\frac{1}{12}\)

A random variable X has the following probability distribution

X	0	1	2	3	4	5
P(X=x)	\(\frac{1}{4}\)	2a	3a	4a	5a	\(\frac{1}{4}\)

Then P(1≤X≤4) is

(a)

\(\frac{10}{21}\)

(b)

\(\frac{2}{7}\)

(c)

\(\frac{1}{14}\)

(d)

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

A random variable X has the following probability mass function

X	-2	3	1
P(X=x)	\(\frac{\lambda}{6}\)	\(\frac{\lambda}{4}\)	\(\frac{\lambda}{12}\)

Then λ is

(a)

1

(b)

2

(c)

3

(d)

4

X is a discrete random variable. Which take values 0,1,2 and P(X = 0) = \(\frac{144}{169}\), P(X = 1) = \(\frac{1}{169}\), then the value of P(X = 2) is ________

(a)

\(\frac{145}{169}\)

(b)

\(\frac{24}{169}\)

(c)

\(\frac{2}{169}\)

(d)

\(\frac{143}{169}\)

Given E(X + c) = 8 and E(X - c) = 12, then the value of c is ___________

(a)

-2

(b)

4

(c)

-4

(d)

2

X is a random variable. Taking the values 3, 4 and 12 with probabilities \(\frac{1}{3},\frac{1}{4}\)and \(\frac{5}{12}\).Then E(X) is

(a)

5

(b)

7

(c)

6

(d)

3

Variance of the random variable. X is 4, Its mean is 2. Then E(X²) is _________

(a)

2

(b)

4

(c)

6

(d)

8

μ₂ = 20, μ¹₂ = 276 for a discrete random variable X. Then the mean of the random variable. X is _________

(a)

16

(b)

5

(c)

2

(d)

1

V(4X + 3) is _________

(a)

7

(b)

16V(X)

(c)

4V(X)

(d)

19

If E(X)  = \(\frac{1}{2}, E(X^2)=\frac{1}{4}\), then V(X) is ________

(a)

0

(b)

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

(c)

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

(d)

1

If f(x) = cx², 0

(a)

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

(b)

\(\frac{4}{3}\)

(c)

\(\frac{8}{3}\)

(d)

\(\frac{3}{8}\)

A probability distribution function is defined by \(F(x)\begin{cases} 0,\ x<0 \\ 1-{ e }^{ -x },\ x\ge 0 \end{cases}\) The probability density function is __________

(a)

\(f(x)\begin{cases} { 3e }^{ -3x },\quad x\ge 0 \\ 0,\quad x<0 \end{cases}\)

(b)

\(\\ f(x)\begin{cases} { 1-e }^{ -3x },\quad x\ge 0 \\ 0,\quad x<0 \end{cases}\)

(c)

\(f(X)=0\forall x\)

(d)

f(x) = 3e - 3x\(\infty\)\(\infty\)

If \(f(x)=\left\{\begin{array}{cc} \frac{A}{x}, & 1<x<e^{3} \\ 0, & \text { otherwise } \end{array}\right.\) is a p.d.f. of a continuous random variable. X then P(X≥e)

(a)

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

(b)

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

(c)

\(\frac{1}{6}\)

(d)

\(\frac{1}{8}\)

The random variable. X has variance 4 and E(X²) = 8. Then the mean of X is __________

(a)

2\(\sqrt{3}\)

(b)

4

(c)

2

(d)

√2

If the p.d.f of a continuous random variable. X is \(f(x)=\left\{\begin{array}{l} \frac{x}{2}, 0 then E(3X²-2X) = _________

(a)

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

(b)

\(\frac{4}{3}\)

(c)

\(\frac{10}{3}\)

(d)

\(\frac{7}{3}\)