Probability Distribution of Random Variable

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.

So, basically, the probability distribution gives the possibility of each outcome of a random experiment or events. It provides the probabilities of different possible occurrence.

This distribution could be defined with any random experiments, whose outcome is not sure or could not be predicted. Let us discuss probability distribution definition, probability distribution function, probability distribution formula and types of probability distribution here, along with how to create probability distribution table.

Probability Distribution Definition

As discussed, 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 set 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 distribution table in terms of variable and probabilities.

Type of Probability Distribution

There are basically two types of probability distribution:

1. Normal Probability Distribution

2. Binomial Probability Distribution

Let us learn both the types of probability distribution one by one;

Normal Probability Distribution

This is also known as a continuous 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 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 probability distribution table.

Normal Probability Distribution is described by probability density function. The formula for this distribution is;

p(x)= 1/\(\sqrt{2\pi \sigma ^2}\).\(e^{(x-\mu )^2/2\sigma ^2}\)


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 Probability Distribution

The binomial Probability 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 probability distribution is;




n = Total number of events

r = Total number of successful events.

p = Success on a single trial probability.

nCr = \(\frac{n!}{r!(n-r)!}\)

1 – p = Failure Probability

We have discussed the probability distribution formula based on its types.

Probability Distribution Function

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

In the case of Normal probability distribution, which is also called a cumulative probability distribution, the probability distribution function of a real-valued random variable X is the function given by;

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

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

Probability Distribution

Probability Distribution Graph

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

P(a<X\(\leq\)b) = FX(b) – FX(a)

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

FX(x) = \(\int_{-\infty }^{x}\) fX(t) dt.

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

P(X=b) = FX(b) – \(\lim_{x\rightarrow b^-}\)

In the case of Binomial probability distribution or a discrete probability distribution, we know the probability distribution 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 : \(S\rightarrow A\)

And A &‌#8838; R, where R is a discrete random variable.

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

fX(x) = Pr (X=x) = P ({s &‌#8712; S : X(s) = x})

Probability Distribution Table

The table for the probability distribution 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. The table formed with respect to the random variable and the probability distribution is;

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?

Answer: 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


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The value of (3 + 4i) (6 - 7i) is