Probability Distributions

Probability Distribution: 

    Probability distribution yields the possible outcomes for any random event. It is also defined based on the underlying sample space as a set of possible outcomes of any random experiment. These settings could be a set of real numbers or a set of vectors or a 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 variables and probabilities.

Random Variables :

     A   random variables is a real-valued function defined on the sample space of a random experiment . In other word , the domain of a random variable is the sample space of a random experiment , while its co-domain is the set of real numbers.

Thus  X: S → R  is a random variable.

Two random variables with equal probability distribution can yet vary with respect to their relationships with other random variables or whether they are independent of these. The recognition of a random variable, which means, the outcomes of randomly choosing values as per the variable’s probability distribution function, are called random variates.

A random variable is usually denoted by a capital letter , like X or Y. A particular value taken by the random variable is usually denoted by the small letter like x .

Note : x is always a real number and the set of all possible outcomes corresponding to a particular value x of X is denoted by the event [X = x].

For example , in the experiment of three seeds , the random variable X taken possible values, namely 0, 1, 2, 3.  The four events are then defined as follows.

           [X = 0] = {NNN},

           [X = 1] = {YNN, NYN, NNY},

           [X = 2] = {YYN, YNY, NYY},

           [X = 3] = {YYY}.

The sample space in this experiment is finite and so is the random variable defined on it .

Types of Random Variable :

There are two types of random variables,

1)   Discrete Random Variable and  

2)   Continuous Random Variable

Discrete Random Variable :

    A random variable is said to be discrete random variable if the number of its possible values is finite or countably infinite .

   The vale of discrete random variable are usually denoted by non-negative integers , that is {0, 1, 2,...}

Example of discrete random variable include the number of children in a family, the number of patients in a hospital ward, the number of cars sold by a dealer , number of stars in the sky and so on. 

Note: The value of a discrete random variavle are obtained by counting.

Continuous Random Variable :

  A random variable is said to be a continuous random variable if the possible values of this random variable form an interval of real numbers.

A continuous random variable has uncountably infinte possible values and these values form an interval of real numbers.

Example of continuous random variable include height of trees ina forest, eeight of students in a class , daily temperature of a city , speed of a vahicle, and so on .

The value of continuous random variable is obtained by measurment. This value can be measured to any degree of accuracy, depending on the unit of measurment. This measurment can be reoresented by a point in an interval of real numbers.

Types of Probability Distribution

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

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

Let us discuss now both the types along with their definition, formula and examples.

Cumulative Probability Distribution

The cumulative probability distribution is also known as a continuous probability distribution. In this distribution, the set of possible outcomes can take on values in 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, a set of complex numbers, a set of prime numbers, a set of whole numbers etc. are 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. A probability density function describes it. The formula for the normal distribution is;

Cumulative Probability Distribution formula

Where,

      • 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

Normal Distribution Examples

Since the normal distribution statistics estimates many natural events so well, it has evolved into a standard of recommendation for many probability queries. Some of the examples are:

      • Height of the Population of the world
      • Rolling a dice (once or multiple times)
      • To judge the Intelligent Quotient Level of children in this competitive world
      • Tossing a coin
      • Income distribution in countries economy among poor and rich
      • The sizes of females shoes
      • Weight of newly born babies range
      • Average report of Students based on their performance

Discrete Probability Distribution

A distribution is 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. It is also known as the probability mass function.

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;

Discrete Probability Distribution formula

Where,

      • 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

Binomial Distribution Examples

As we already know, binomial distribution gives the possibility of a different set of outcomes. In the real-life, the concept is used for:

      • To find the number of used and unused materials while manufacturing a product.
      • To take a survey of positive and negative feedback from the people for anything.
      • To check if a particular channel is watched by how many viewers by calculating the survey of YES/NO.
      • The number of men and women working in a company.
      • To count the votes for a candidate in an election and many more.

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)

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,

Probability distribution 1

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

Cumulative dstribution function for random variable

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}).


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