Random variables and its probability distributions: A variable that is used to quantify the outcome of a random experiment is a random variable. Since there are two forms of data, discrete and continuous, there are two types of random variables. A discrete random variable can have an exact value, whereas the value of a continuous random variable will lie within a specific range.

Exponential and normal random variables are the types of continuous random variables, while binomial, Poison’s, Bernoulli’s, and geometric are the types of discrete random variables. Probability distribution is a function that calculates the likelihood of all possible values for a random variable.

Definition of Random Variable

A random variable is a type of variable whose value is determined by the numerical results of a random experiment. A random variable is also called a stochastic variable. As random variables must be quantifiable, they are always real numbers.

Random Variable vs Algebraic Variable

A random variable can have different values because a random event might have multiple outcomes. As a result, do not even confuse a random variable with an algebraic variable. In an algebraic equation, an algebraic variable represents the value of an unknown quantity. On the other hand, a random variable can have a collection of values that could be the result of a random experiment.

Example: Assume two dice are rolled, and the random variable \(X\) represents the sum of the numbers. The smallest value of \(X\) will be \(2\) , while the largest possible value is \(12\) . As a result, \(X\) may be any number equal to or between \(2\) and \(12\) . After assigning probabilities to each outcome, the probability distribution of \(X\) may be calculated.

Properties of Random Variable

• Random variables take only positive real values
• If \(C\) is any real number and \(X\) is any random variable, then \(CX\) is a random variable.
• If \(x\) and \(y\) are two random variables, then
  (i) \(x+y\) is a random variable
  (ii) \(xy\) is a random variable

• For any constants \(\mathrm{K}\) and \(\mathrm{C}, \mathrm{K} x+\mathrm{C} y\) is also a random variable.
• The modulus \((|X|)\) is also a random variable for any random variable \(\mathrm{X}\).

Types of Random Variables

Random variables can be categorised based on the available data type, as shown below.

Continuous Random Variable

A continuous random variable is that which has infinite possible values. A variable like this is defined over a range of values rather than a single value. The weight of a person is an example of a continuous random variable. A probability density function is used to describe a continuous random variable because the probability that it will take on an exact value is zero.

Types of Continuous Random Variables

Exponential Continuous Random Variable

Exponential random variable	\(\mathrm{X} \sim \operatorname{Exp}(\lambda)\)
The probability density function of the exponential random variable	\(f(x) = \left\{ {\begin{array}{*{20}{c}}{\lambda {e^{ – \lambda x}},}&{x \ge 0}\\{0,}&{x < 0}\end{array}} \right\}\)

Normal Continuous Random Variable

Normal random variable	\(\mathrm{X} \sim\left(\mu, \sigma^{2}\right)\)
Probability density function	\(f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}\)

Discrete Random Variable

A discrete random variable can have a finite number of different values. For example, a discrete random variable can be used to represent the number of children in a family. A probability distribution is used to determine what values a random variable can take and how frequently it does so. 

Types of Discrete Random Variables

Binomial Discrete Random Variable

A binomial random variable indicates the number of successes in a binomial experiment. A binomial experiment consists of a set number of repeated Bernoulli trials with only two possible outcomes: success or failure. The number of trials is denoted by \(n\), while the chance of success is denoted by \(p\).

Binomial random variable	\(\mathrm{X} \sim \operatorname{Bin}(n, p)\)
Probability density function	\(P\left( {X = x} \right) = \left( {\begin{array}{*{20}{c}} n \\ x \end{array}} \right){p^x}{\left( {1 – p} \right)^{n – x}}\)

Geometric Discrete Random Variable

In a Bernoulli trial, the probability of success is \(p\), and the probability of failure is \(1-p\).

Geometric random variable	\(\mathrm{X} \sim \mathrm{G}(\mathrm{p})\)
Probability density function	\(P(X=x)=(1-p)^{x-1} p\)

Bernoulli’s Discrete Random Variable

The simplest sort of random variable is Bernoulli’s random variable. There are only two possible values for this variable: \(1\) for success and \(0\) for failure.

Bernoulli’s random variable	\(X \sim\) Bernoulli \((p)\)
Probability density function	\(P(X = x) = \left\{ {\begin{array}{*{20}{c}}{p,}&{{\text{if}}\,\,x = 1}\\{1 – p,}&{{\text{if}}\,\,x = 0}\end{array}} \right\}\)

Poisson Discrete Random Variable

A Poisson random variable illustrates how many times an event will happen in the given time. These events occur at a consistent rate and in random order.

Poisson discrete random variable	\(X \sim\) Poisson \((\lambda)\)
Probability density function	\(P(X=x)=\frac{\lambda^{x} e^{-\lambda}}{x !}\)

Mean of a Random Variable

The weighted average of all the values of a random variable can also be described as the mean or expected value of the variable. It is represented by \(E[X]\). Here, \(X\) is the random variable. The following are the formulas for calculating the mean of a random variable:

Mean of discrete random variable	\(E\left[ X \right] = \sum {xP\left( {X = x} \right)}\) where \({P\left( {X = x} \right)}\) is the probability mass function.
Mean of continuous random variable	\(E\left[ X \right] = \int {xf\left( x \right)dx}\) where \(f\left( x \right)\)  is the probability density function

Variance of a Random Variable

Variance of a random variable is the expected value of the square of the difference between the random variable and the mean. Var\([X]\) or \(\sigma^{2}\) represents the variance of a random variable. If \(\mu\) is the mean, then the variance can be calculated as follows:

Variance of continuous random variable	\(\operatorname{Var}[\mathrm{X}]=\int(\mathrm{x}-\mu)^{2} \mathrm{f}(\mathrm{x}) \mathrm{dx}\)
Variance of discrete random variable	\(\operatorname{Var}[\mathrm{X}]=\sum(\mathrm{x}-\mu)^{2} \mathrm{P}(\mathrm{X}=\mathrm{x})\)

Definition of Probability Distribution

A probability distribution is a function that calculates the likelihood of all possible values for a random variable. A probability distribution and probability mass functions can both be used to define a discrete probability distribution. A continuous probability distribution is described using a probability distribution function and a probability density function.

Probability Distribution Function

The probability distribution function is also known as the cumulative distribution function (CDF). This gives the likelihood of a random variable, \(\mathrm{X}\). When evaluated at a point, \(x\), it takes values less than or equal to \(x\).

\(F(x)=P(X \leq x)\)

Furthermore, if \((a, b]\) is a semi-closed interval, the probability distribution function is provided by the formula given below.

\(\mathrm{P}(\mathrm{a}<\mathrm{X} \leq \mathrm{b})=\mathrm{F}(\mathrm{b})-\mathrm{F}(\mathrm{a})\)

A random variables probability distribution function is always between \(0\) and \(1\) . It is a function that does not decrease.

Formulas of Probability Distribution

A probability distribution has multiple formulas depending on the type of distribution a random variable follows.

Probability Distribution of a Discrete Random Variable

For a discrete random variable, the formulas for the probability distribution function and the probability mass function are as follows:

Probability distribution function	\(\mathrm{F}(\mathrm{x})=\mathrm{P}(\mathrm{X} \leq \mathrm{x})\)
Probability mass function	\(\mathrm{p}(\mathrm{x})=\mathrm{P}(\mathrm{X}=\mathrm{x})\)

Probability Distribution of a Continuous Random Variables

We cannot use the probability mass function to characterise such distribution since the likelihood that a continuous random variable would take on an exact value is \(0\) . Hence, we use the probability density function.

Probability distribution function	\(F(x)=P(X \leq x)\)
Probability mass function	\(\mathrm{f}(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{F}(\mathrm{x}))\), where \({\rm{F}}({\rm{x}}) = \int_{ – \infty }^x f (u)du\)

Solved Examples of Random Variables and its Probability Distributions

Here are a few solved examples for the students,

Q.1. If two dice are rolled, what is the probability distribution of the sum of the dice?
Sol: 
Possible outcomes \(=(2,3,4,5,6,7,8,9,10,11,12)\)
Assume \(1\) is rolled on the first die and \(1\) is rolled on the second die.
The total will then be \(2\) , because no alternative set of integers can produce the same result.
Probability of having the result is \(2=\frac{1}{36}\).
Other numbers are treated in the same way.

\(x\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)	\(12\)
\(P(x)\)	\(\frac{1}{36}\)	\(\frac{2}{36}\)	\(\frac{3}{36}\)	\(\frac{4}{36}\)	\(\frac{5}{36}\)	\(\frac{6}{36}\)	\(\frac{5}{36}\)	\(\frac{4}{36}\)	\(\frac{3}{36}\)	\(\frac{2}{36}\)	\(\frac{1}{36}\)

Q.2. Find the probability that a die will show a number less than \(6\), if rolled multiple times.
Sol:
Possible outcomes are \(\left\{ {1,2,3,4,5,6} \right\}\)
Numbers less than \(6 = \left\{ {1,2,3,4,5} \right\}\)
\(\mathrm{P}(\mathrm{X}<6)=\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)+\mathrm{P}(\mathrm{X}=3)+\mathrm{P}(\mathrm{X}=4)+\mathrm{P}(\mathrm{X}=5)\)
\(P(X<6)=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{5}{6}\)
\(\therefore \mathrm{P}(\mathrm{X}<6)=\frac{5}{6}\)

Q.3. The probability mass function is given by \(p\left( x \right) = \left\{ {\begin{array}{*{20}{c}} \begin{gathered} 0.2, \hfill \\ 0.3, \hfill \\ 0.1, \hfill \\ 0.4, \hfill \\ 0, \hfill \\ \end{gathered} &\begin{gathered} x = 0.1 \hfill \\ x = 0.2 \hfill \\ x = 0.3 \hfill \\ x = 0.4 \hfill \\ {\text{otherwise}} \hfill \\ \end{gathered} \end{array}} \right.\) for a discrete variable \(x.\) Find the value of \(P(x<4)\)
Sol:
Given : \(p\left( x \right) = \left\{ {\begin{array}{*{20}{c}} \begin{gathered} 0.2, \hfill \\ 0.3, \hfill \\ 0.1, \hfill \\ 0.4, \hfill \\ 0, \hfill \\ \end{gathered} &\begin{gathered} x = 0.1 \hfill \\ x = 0.2 \hfill \\ x = 0.3 \hfill \\ x = 0.4 \hfill \\ {\text{otherwise}} \hfill \\ \end{gathered} \end{array}} \right.\)
The value of \(P(x<4)=P(X=1)+P(X=2)+P(X=3)\)
\(P(x<4)=0.2+0.3+0.1\)
\(\therefore P(x<4)=0.6\)

Q.4. Assume that you have a \(25 \%\) chance of hitting the bullseye in a game of darts. What are the chances of hitting the bullseye five times if you take a total of \(15\) shots?
Sol:
Given \(n=15, x=5\) and \(P=25 \%=\frac{25}{100}=\frac{1}{4}\)
The binomial probability distribution function is given by
\(P\left( {X = x} \right) = \left( {\begin{array}{*{20}{c}} n \\ x \end{array}} \right){p^x}{\left( {1 – p} \right)^{n – x}}\)
\(P\left( {X = 5} \right) = \left( {\begin{array}{*{20}{c}} {15} \\ 5 \end{array}} \right){0.25^5}{\left( {1 – 0.25} \right)^{15 – 5}}\)
\( = \left( {\begin{array}{*{20}{c}} {15} \\ 5 \end{array}} \right){0.25^5}{\left( {0.75} \right)^{10}}\)
\(\therefore P(X=5)=0.165\)

Q.5. Find the mean of the following distribution function \(F\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\left( {p + 1} \right){x^{p + 1}},}&{0 \leqslant x \leqslant 1} \\ {0,}&{{\text{otherwise}}} \end{array}} \right.\)
Sol:
Given: \(p>-1\)
The mean of the distribution is given by \(\mathrm{E}(\mathrm{x})=\int x . f(x) d x\)
So, \(E(x)=\int_{0}^{1}(p+1) x^{p+1} d x\)
\(E(x)=\left[\frac{(p+1) x^{p+2}}{p+2}\right]_{0}^{1}\)
\(\therefore E(x)=\frac{p+1}{p+2}\)

Summary

A random variable is used to quantify a random experiment’s outcome. There are two forms of data, discrete and continuous. Hence, there are two types of random variables. A discrete random variable can have an exact value, whereas the value of a continuous random variable will lie within a specific range.

Exponential and normal are the types of continuous random variables. Geometric, binomial, and Bernoulli are the types of discrete random variables. A probability distribution is a function that calculates the likelihood of all possible values for a random variable. Probability distributions are diagrams that depict how probabilities are spread throughout the values of a random variable.

FAQs of Random Variables and its Probability Distributions

Students must have many questions with respect to Random Variables and its Probability Distributions. Here are a few commonly asked questions and answers.

Q.1. What is meant by random variable?
Ans: A random variable is that which represents all possible outcomes of a random event.

Q.2. What are two types of random variables?
Ans: Random variables are of two types: discrete random variables and continuous random variables. A discrete random variable can have a single value, while a continuous random variable has a range of values.

Q.3. What is a probability distribution?
Ans: The probability that a random variable will take on a specific value is represented by a probability distribution.

Q.4. What are the types of probability distributions?
Ans: The various types of probability distributions include binomial, Bernoulli’s, normal, and geometric distributions.

Q.5. What is the difference between discrete and continuous random variables?
Ans: A discrete random variable can have an exact value, whereas a continuous random variable’s value will lie within a specific range.

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