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

Where,

The real numbers x₁ , x₂ ,..., x_n are the possible values of the random variable X and p_i (i = 1,2,..., n) is the probability of the random variable X taking the value xi i.e., P(X = x_i ) = p_i

Mean of a Random Variable

Let X be a random variable whose possible values x₁ , x₂ ,..., x_n occur with probabilities p₁ , p₂ , p₃ ,..., p_n, respectively. The mean of X, denoted by μ, is the number   i.e. the mean of X is the weighted average of the possible values of X, each value 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,     

Variance of a random variable

Let X be a random variable whose possible values x₁ , x₂ ,...,x_n occur with probabilities p(x₁ ), p(x₂ ),..., p(x_n ) respectively.

Let μ = E (X) be the mean of X. The variance of X, denoted by Var (X) or  is defined as

And the non-negative number

is called the standard deviation of the random variable X.