\(\text { If } \ p(x) \ = \begin{cases}\frac{x}{20}, & x=0,1,2,3,4,5 \\ 0, & \text { otherwise }\end{cases}\)
Find
(i) P(X<3) and 
(ii) P(2\(\leq\)4)

If you toss a fair coin three times, the outcome of an experiment consider as random variable which counts the number of heads on the upturned faces. Find out the probability mass function and check the properties of the probability mass function.

Two unbiased dice are thrown simultaneously and sum of the upturned faces considered as random variable. Construct a probability mass function.

A coin is tossed thrice. Let X be the number of observed heads. Find the cumulative distribution function of X.

A continuous random variable X has the following p.d.f f(x) = ax, 0\(\le\)x\(\le\)1
Determine the constant a and also find P\(\\ \left[ X\le \frac { 1 }{ 2 } \right] \)

Let X be a discrete random variable with the following p.m.f
\(p(x) = \begin{cases}0.3 & \text { for } x =3 \\ 0.2, & \text { for } x = 5 \\ 0.3, & \text { for } x = 8 \\ 0.2, & \text { for} x = 10 \\ 0, & \text { otherwise } \\ \end{cases}\)
Find and plot the c.d.f. of X.

The discrete random variable X has the following probability function \(P(X=x) = \begin{cases}kx & x =2, 4, 6 \\ k(x - 2), & x = 8 \\ 0, & \text { otherwise } \\ \end{cases}\) where k is a constant. Show that k = \(\frac{1}{18}\)

Explain the terms
(i) probability mass function,
(ii) probability density function and
(iii) probability distribution function.

What are the properties of
(i) discrete random variable and
(ii) continuous random variable?

State the properties of distribution function.

An urn contains four balls of red, black, green and blue colours. There is an equal probability of getting any coloured ball. What is the expected value of getting a blue ball out of 30 experiments with replacement?

A fair die is thrown. Find out the expected value of its outcomes.

Suppose the probability mass function of the discrete random variable is

X=x	0	1	2	3
p(x)	0.2	0.1	0.4	0.3

What is the value of E(3X + 2X²) ?

Consider a random variable X with probability density function \(f(x)= \begin{cases}4x^3 & \text { if } 0< x < 1 \\ 0, & \text { otherwise }\end{cases}\)
Find E(X) and V(X).

If f (x) is defined by f(x)=ke^-2x,  0\(\le\)x<\(\infty\) is a density function. Determine the constant k and also find mean.

The time to failure in thousands of hours of an important piece of electronic equipment used in a manufactured DVD player has the density function.\(f(x)= \begin{cases}3e^{-3x} & x > 0 \\ 0, & \text { otherwise }\end{cases}\)
Find the expected life of the piece of equipment.

A commuter train arrives punctually at a station every 25 minutes. Each morning, a commuter leaves his house and casually walks to the train station. Let X denote the amount of time, in minutes, that commuter waits for the train from the time he reaches the train station. It is known that the probability density function of X is
\(f(x)= \begin{cases}\frac{1}{25}, \text { for } & 0 < x < 25 \\ 0, & \text { otherwise }\end{cases}\)
Obtain and interpret the expected value of the random variable X.

The following table is describing about the probability mass function of the random variable X

x	3	4	5
P(x)	0.1	0.1	0.2

Find the standard deviation of x.

Let X be a continuous random variable with probability density function
\(f(x)=\left\{\begin{array}{l} \frac{3}{x^{4}}, x \geq 1 \\ 0, \text { otherwise } \end{array}\right.\)
Find the mean and variance of X.

In a business venture a man can make a profit of Rs. 2,000 with a probability of 0.4 or have a loss of Rs. 1,000 with a probability of 0.6. What is his expected, variance and standard deviation of profit?

The number of miles an automobile tire lasts before it reaches a critical point in tread wear can be represented by a p.d.f.
\(f(x)= \begin{cases}\frac{1}{30} e^{-\frac{x}{30}}, & \text { for } x>0 \\ 0, & \text { for } x \leq 0\end{cases}\)
Find the expected number of miles (in thousands) a tire would last until it reaches the critical tread wear point.

A person tosses a coin and is to receive Rs. 4 for a head and is to pay Rs. 2 for a tail. Find the expectation and variance of his gains.

The p.d.f. of X is defined as \(f(x)= \begin{cases}k & \text { for } 0< x \leq 4 \\ 0, & \text { otherwise }\end{cases}\) Find the value of k and also find  P(2\(\le\)X\(\le\)4).

The probability distribution function of a discrete random variable X is
\(F(x)=\left\{\begin{array}{l} 2k, x = 1 \\ 3k, x = 3 \\ 4k, x = 5 \\ 0, \text{otherwise} \end{array}\right.\)
where k is some constant. Find (a) k and (b) P(X>2).

Consider a random variable X with p.d.f
\(f(x)=\left\{\begin{array}{l} 3 x^{2}, \text { if } 0< x< 1 \\ 0, \text { otherwise } \end{array}\right.\)
Find E(X) and V(3X-2).