Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are

If P(X = 0) = 1 − P(X = 1). If E(X) = 3Var(X), then P(X = 0).

A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
(i) the probability mass function
(ii) the cumulative distribution function
(iii) P(4 ≤ X < 10)
(iv) P(X ≥ 6)

A random variable X has the following probability mass function.

Find
(i) the value of k
(ii) P(2 \(\le\) X < 5)
(iii) P(3 < X )

Find the binomial distribution function for each of the following.
(i) Five fair coins are tossed once and X denotes the number of heads.
(ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
\(\\ \\ \\ \\ \\ f(x)=\begin{cases} \begin{matrix} \frac { { x }^{ 2 }+1 }{ k } & forx=0,1,2 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\\ \\ \\ \\ \\ \\ \)
Find
(i) the value of k
(ii) cumulative distribution function
(iii) P(X ≥ 1).

Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X.

If X is the random variable with distribution function F(x) given by,
\(F(x)=\begin{cases} \begin{matrix} 0 & x<0 \end{matrix} \\ \begin{matrix} x & 0\le x<1 \end{matrix} \\ \begin{matrix} 1 & 1\le x \end{matrix} \end{cases}\) 
then find
(i) the probability density function f(x)
(ii) P(0.2 ≤ X ≤ 0.7)

The probability density function of X is given
\(f(x)=\begin{cases} \begin{matrix} { Ke }^{ \frac { -x }{ 3 } } & \begin{matrix} for & x>0 \end{matrix} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} for & x\le 0 \end{matrix} \end{matrix} \end{cases}\)
Find
(i) the value of k
(ii) the distribution function.
(iii) P(X <3)
(iv) P(5 ≤X)
(v) P(X ≤ 4)

A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is 
\(f(x)= \begin{cases}\frac{1}{30} & 0

A six sided die is marked ‘1’ on one face, ‘2’ on two of its faces, and ‘3’ on remaining three faces. The die is rolled twice. If X denotes the total score in two throws.
(i) Find the probability mass function.
(ii) Find the cumulative distribution function.
(iii) Find P(3 ≤ X< 6)
(iv) Find P(X ≥ 4) .