Probability Distribution of Random Variable

The probability distribution gives the possibility of each outcome of a random experiment or events. It provides the probabilities of different possible occurrence. To recall, probability is a measure of uncertainty of various phenomenon. Like, if you throw a dice, what is the possible outcomes of it, is defined by the probability. This distribution could be defined with any random experiments, whose outcome is not sure or could not be predicted.

Probability Distribution Definition

Probability distribution yields the possible outcomes for any random event. It is also defined on the basis of underlying sample space as a set of possible outcomes of any random experiment. These settings could be a set of real numbers or set of vectors or set of any entities. it is a part of probability and statistics.

Random experiments are defined as the result of an experiment, whose outcome cannot be predicted. Suppose, if we toss a coin, we cannot predict, what outcome it will appear, either it will come as Head or as Tail. The possible result of a random experiment is called an outcome. And the set of outcomes is called a sample point. With the help of these experiments or events, we can always create a probability pattern table in terms of variable and probabilities.

Types of Probability Distribution

There are basically two types of probability distribution which are used for different purposes and various types of data generation process.

  1. Normal or Cumulative Probability Distribution
  2. Binomial or Discrete Probability Distribution

Let us discuss now both the type along with its definition and formula.

Normal / Cumulative Probability Distribution

This is also known as a continuous or cumulative probability distribution. In this distribution, the set of possible outcomes can take on values on a continuous range.

For example, a set of real numbers, is a continuous or normal distribution, as it gives all the possible outcomes of real numbers. Similarly, set of complex numbers, set of a prime number, set of whole numbers etc are the examples of Normal Probability distribution. Also, in real-life scenarios, the temperature of the day is an example of continuous probability. Based on these outcomes we can create a distribution table. It is described by a probability density function. The formula for the normal distribution is:

Normal Probability Distribution Function Formula


  • μ = Mean Value
  • σ = Standard Distribution of probability.
  • If mean(μ) = 0 and standard deviation(σ) = 1, then this distribution is known to be normal distribution.
  • x = Normal random variable

Binomial / Discrete Probability Distribution

This distribution is also called a discrete probability distribution, where the set of outcomes are discrete in nature.

For example, if a dice is rolled, then all the possible outcomes are discrete and give a mass of outcomes. This is also known as probability mass functions.

So, the outcomes of binomial distribution consist of n repeated trials and the outcome may or may not occur. The formula for the binomial distribution is;

Binomial Probability Distribution Formula


  • n = Total number of events
  • r = Total number of successful events.
  • p = Success on a single trial probability.
  • nCr = [n!/r!(n−r)]!
  • 1 – p = Failure Probability

Probability Distribution Function

A function which is used to define the distribution of a probability is called a Probability distribution function. Depending upon the types, we can define these functions. Also, these functions are used in terms of probability density functions for any given random variable.

In the case of Normal distribution,  the function of a real-valued random variable X is the function given by;

FX(x) = P(X ≤ x)

Where P shows the probability that the random variable X occurs on less than or equal to the value of x.

For a closed interval, (a→b), the cumulative probability function can be defined as;

P(a<X ≤ b) = FX(b) – FX(a)

If we express, the cumulative probability function as integral of its probability density function fX , then,

In the case of a random variable X=b, we can define cumulative probability function as;

In the case of Binomial distribution, as we know it is defined as the probability of mass or discrete random variable gives exactly some value. This distribution is also called probability mass distribution and the function associated with it is called a probability mass function.

Probability mass function is basically defined for scalar or multivariate random variables whose domain is variant or discrete. Let us discuss its formula:

Suppose a random variable X and sample space S is defined as;

X : S → A

And A ∈ R, where R is a discrete random variable.

Then the probability mass function fX : A → [0,1] for X can be defined as;

fX(x) = Pr (X=x) = P ({s ∈ S : X(s) = x})

Probability Distribution Table

The table could be created on the basis of a random variable and possible outcomes. Say, a random variable X is a real-valued function whose domain is the sample space of a random experiment. The probability distribution P(X) of a random variable X is the system of numbers.

X X1 X2 X3 ………….. Xn
P(X) P1 P2 P3 …………… Pn

where Pi > 0, i=1 to n and P1+P2+P3+ …….. +Pn =1

Probability Distribution Example and Solution

Example: A coin is tossed twice. X is the random variable of the number of heads obtained. What is the probability distribution of x?


First write, the value of X= 0, 1 and 2, as the possibility are there that

No head comes

One head and one tail comes

And head comes in both the coins

Now the probability distribution could be written as;

P(X=0) = P(Tail+Tail) = ½ * ½ = ¼

P(X=1) = P(Head+Tail) or P(Tail+Head) = ½ * ½ + ½ *½ = ½

P(X=2) = P(Head+Head) = ½ * ½ = ¼

We can put these values in tabular form;

X 0 1 2
P(X) 1/4 1/2 1/4