Random Variables and its Probability Distributions - Practice Questions & MCQ

A random variable is a real valued function whose domain is the sample space of a random experiment. It is a numerical description of the outcome of a statistical experiment.

A random variable is usually denoted by X.

For example, consider the experiment of tossing a coin two times in succession. The sample space of the experiment is S = {HH, HT, TH, TT}.

If X is the number of tails obtained, then X is a random variable and for each outcome, its value is given as 

X(TT) = 2, X (HT) = 1, X (TH) = 1, X (HH) = 0

Probability Distribution of a Random Variable

The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable.
The probability distribution of a random variable X is the system of numbers

$
\begin{array}{cccccccc}
X & : & x_1 & x_2 & x_3 & \ldots & \ldots & x_n \\
P(X) & : & p_1 & p_2 & p_3 & \ldots & \ldots & p_n \\
& p_i \neq 0, & \sum_{i=1}^n p_i=1, & i=1,2,3, \ldots n
\end{array}
$

The real numbers $x_1, x_2, \ldots, x_n$ are the possible values of the random variable $X$ and $p_i(i=1,2, \ldots, n)$ is the probability of the random variable $X$ taking the value $x i$ i.e., $P\left(X=x_i\right)=p_i$

Mean of a Random Variable being weighted by its probability with which it occurs.

The mean of a random variable X is also called the expectation of X, denoted by E(X).

Thus,     

$
\text { mean }(\mu)=\frac{\sum_{i=1}^n p_i x_i}{\sum_{i=1}^n p_i}=\sum_{i=1}^n x_i p_i \quad\left(\because \sum_{i=1}^n p_i=1\right)
$

Variance of a random variable
Let $X$ be a random variable whose possible values $x_1, x_2, \ldots, x_n$ occur with probabilities $p\left(x_1\right), p\left(x_2\right), \ldots, p\left(x_n\right)$ respectively.
Let $\mu=E(X)$ be the mean of $X$. The variance of $X$, denoted by $\operatorname{Var}(X)$ or $\sigma_x^2$ is defined as

$
\sigma_x^2=\operatorname{Var}(\mathrm{X})=\sum_{i=1}^n\left(x_i-\mu\right)^2 p\left(x_i\right)
$

And the non-negative number

$
\sigma_x=\sqrt{\operatorname{Var}(\mathrm{X})}=\sqrt{\sum_{i=1}^n\left(x_i-\mu\right)^2 p\left(x_i\right)}
$

is called the standard deviation of the random variable $\mathbf{X}$.