Value which is obtained by multiplying possible values of random variable with probability of occurrence and is equal to weighted average is called ________.

Probability which explains x is equal to or less than particular value is classified as ________.

If X is a discrete random variable and p(x) is the probability of X, then the expected value of this random variable is equal to ________.

Which of the following is not possible in probability distribution?

A discrete probability distribution may be represented by ________.

Construct cumulative distribution function for the given probability distribution.

The discrete random variable X has the probability function

Show that k = 0.1.

Two coins are tossed simultaneously. Getting a head is termed as success. Find the probability distribution of the number of successes.

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

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 p.d.f
f(x) = 5x⁴, 0\(\le\)x\(\le\)1 
Find a₁ and a₂ such that
i) P[X\(\le\)a₁] = P[X>a₁]   
ii) P[X>a₂] = 0.05

Determine the mean and variance of a discrete random variable, given its distribution as follows.

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.