Random Variables and its Probability Distributions MCQ

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Random Variables and its Probability Distributions

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).

$\begin{array}{|c|c|c|} \hline \mathrm{Random\;variable\;\;\left (x_i \right )} & \mathrm{Probability\;\;\left (p_i \right )} & \mathrm{p_ix_i} \\ \hline \mathrm{x_{1 }} & {\mathrm{p}_{1}} & {\mathrm{p}_{1} \mathrm{x}_{1}} \\ \hline \mathrm{x_{2 }} & {\mathrm{p}_{2}} & {\mathrm{p}_{2} \mathrm{x}_{2}} \\ \hline \mathrm{x_{3 }} & {\mathrm{p}_{3}} & {\mathrm{p}_{3} \mathrm{x}_{3}} \\ \hline \ldots & \ldots & \ldots \\ \hline \ldots & \ldots & \ldots \\ \hline \mathrm{x_{n}} & {\mathrm{p}_{n}} & {\mathrm{p}_{n} \mathrm{x}_{n}} \\ \hline\end{array}$

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 $\mathbf{X}$.