Random Variables and its Probability Distributions

Following is the distribution function $F\left(x\right)$ of a discrete random variable $X$

$\begin{array}{|lllllll|}\hline x& 1& 2& 3& 4& 5& 6\\ \mathrm{F}\left(x\right)& 0.2& 0.37& 0.48& 0.62& 0.85& 1\\ \hline\end{array}$

Find $P(X\le 5|X>3).$

In a game, a man wins Rs. $100$ if he gets $5$ or $6$ on a throw of a fair die and loses Rs. $50$ for getting any other number on the die. If he decides to throw the die either till he gets a $5$ or a $6$ or to a maximum of three throws, then his expected gain/loss (in rupees) is:

Three balanced coins are tossed simultaneously. If $X$ denotes the number of heads, find the probability distribution of $X$.

Product of the variance and standard deviation of the random variable X whose probability
distribution is given below

Let the p.m.f. of a random variable $X$ be $\underset{}{P\left(x\right)}=\frac{3-x}{10} \text{for } x=-1,0,1,2$ and $0$ otherwise. Then value of $E\left(X\right)$ is

If $x$ is a random variable with probability distribution $p\left(x=k\right)=\frac{(k+1)C}{2^k}, k=0, 1, 2, 3,\dots \dots \dots .,$ then $C=$

If the mean and variance of a binomial variable $X$ are $2.4$ and $1.44$ respectively, find the parameter $n$ of the distribution $X$. (Binomial).

A cubical die is thrown. Find the mean and variance of $x$, giving the number on the face shows up.

The mean and variance of a binomial distribution are 4 and 3 respectively. Fix the distribution and find $P\left(X\ge 1\right)$.

Following is the distribution function F (x) of a discrete r.v. X

$\begin{array}{|lllllll|}\hline x& 1& 2& 3& 4& 5& 6\\ \mathrm{F}\left(x\right)& 0.2& 0.37& 0.48& 0.62& 0.85& 1\\ \hline\end{array}$

Find $P(X\le 5|X>3).$

Let $X$ have p.m.f. $p\left(x\right)=k·x^2;x=1,2,3,4$

$=0;$ otherwise.

Find variance of $X$.

The p.m.f. for $X=$ number of major defects in a randomly selected appliance of a certain type is:

Find the standard deviation of $X$. 

A r.v. $X$ has the following probability distribution :

Find the value of $k$ and calculate mean and variance of $X$. 

The p.m.f. of a r.v. $X$ is

$P\left(x\right)=\frac{1}{15},$ for $x=1,2,\dots ,14,15$
        $=0$                otherwise.

Find $Var\left(X\right)$.

The probability distribution of a discrete random variable $X$ is

Determine the value of $k$.

A random variable $X$, has the probability distribution as given below. Let $E\mathit{=}\left\{X|X\mathrm{is} \mathrm{prime} \mathrm{number}\right\}$ and $F=\left\{X\mid X<4\right\},$ then $P\left(E\cup F\right)=$

Following is the distribution function $F\left(x\right)$ of a discrete r.v. $X$

$\begin{array}{|lllllll|}\hline x& 1& 2& 3& 4& 5& 6\\ \mathrm{F}\left(x\right)& 0.2& 0.37& 0.48& 0.62& 0.85& 1\\ \hline\end{array}$

Find the probability distribution of $X$.

A random variable $X$ has the following probability distribution

Find $p(x>2)+p(0<x<4)$.

Obtain the probability distribution of the number of sixes in two tosses of a fair die.

Determine $k$ such that the following function is a probability mass function.

$P\left(x\right)=\left\{\begin{array}{l}k\left(\begin{array}{c}4\\ x\end{array}\right), x=0,1,2,3,4, k>0\\ 0, \mathrm{otherwise}\end{array}\right.$