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 ________.

A variable that can assume any possible value between two points is called ________.

If c is a constant, then E(c) is ________.

E[X-E(X)] is equal to ________.

\(\int _{ -\infty }^{ \infty }{ f(x)dx } \) is always equal to ________.

If p(x) =\(\frac{1}{10}\), c = 10, then E(X) is ________.

In a discrete probability distribution the sum of all the probabilities is always equal to ________.

The probability density function p(x) cannot exceed ________.

The distribution function F(x) is equal to ________.

Construct cumulative distribution function for the given probability distribution.

The discrete random variable X has the probability function

Show that k = 0.1.

The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as \( f(x)\begin{cases} { Ae }^{ -x/5 },\quad \text{for}\quad x\ge 0 \\ 0 \quad ,\quad \text{otherwise }\end{cases}\)
(a) Find the value of A that makes fix) a p.d.f,
(b) What is the probability that the number of minutes that person will talk over the phone is
(i) more than 10 minutes
(ii) less than 5 minutes and
(iii) between 5 and 10 minutes.

Define random variable.

Distinguish between discrete and continuous random variable.

State the properties of distribution function.

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

Find the standard deviation of x.

What do you understand by Mathematical expectation?

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 number of cars in a household is given below.

Estimate the probability mass function. Verify p(x_i ) is a probability mass function.

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.

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.