Determine whether the following is a probability distribution of a random variable X.

X	0	1	2
P(X)	0.6	0.1	0.2

An unbiased die is rolled. If the random variable X is defined as
X(w) = {1, the outcome w is an even number    
{0, if the outcome w is an odd number
Find the probability distribution of X.

Two eggs are drawn at random without replacement from a bag containing two bad eggs and eight good eggs. Find the probability of getting two bad eggs?

A random variable X has the probability mass function

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

then find k

A discrete random variable. X has the following probability distribution

X	0	1	2	3	4	5	6	7	8
P(X)	a	3a	5a	7a	9a	11a	13a	15a	17a

Pind the value of a and P(X< 3)

Verify whether \(f(x)=\begin{cases} \frac { 2x }{ 9 } ,\quad 0\le x\le \\ 0,\quad elsewhere \end{cases}\) is a probability density function

A continuous random variable. X has the p.d.f. defined by \(f(x)=\left\{\begin{array}{l} C e^{-a x}, \quad 0<x<\infty \\ 0, \quad \text { elsewhere } \end{array}\right.\) Find the value of C if a> 0

In an entrance examination a student has to answer all the 120 questions. Each question has four options and only one option is correct. A student gets 1 mark for a correct answer and loses \(\frac{1}{2}\) mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the answer to each question at random?

In a gambling game a man wins Rs. 10 if he gets all heads or all tails and loses Rs. 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.

Find the mean for the probability density function \(f(x)=\begin{cases} \frac { 1 }{ 24 } ,-12\le x\le 12 \\ 0,\quad otherwise \end{cases}\)

A continuous random variable X follows the probability law \(f(x)= \begin{cases}k x(1-x)^{10}, & 0. Find k.

If \(\mathrm{F}(x)=\frac{1}{\pi}\left(\frac{\pi}{2}+\tan ^{-1} x\right)-\infty<x<\infty\) distribution function of a continuous variable X, find \(\mathrm{P}(0 \leq x \leq 1)\)

For the probability density function \(f(x)=\left\{\begin{array}{cc} 2 e^{-2 x} & x>0 \\ 0 & x \leq 0 \end{array} .\right.\) Find F(2).

Verify whether f(x)  \(= \begin{cases}\frac{2 x^{\circ}}{9}, & 0 \leq x \leq 3 \\ 0 & \text { elsewhere }\end{cases}\)probability density function.

Verify whether \(f(x)=\frac{1}{\pi} \frac{1}{1+x^{2}},-x is a probability density function.