Probability Distribution

What is the Probability Distribution?

A probability distribution is a table or an equation that interconnects each outcome of a statistical experiment with its probability of occurrence. 

To understand the concept of a probability distribution, it is important to know variables, random variables, and some other notations.

Variables:  A variable is defined as any symbol  that can take any particular set of values.

Random Variable: When the value of any variable is the outcome of a statistical experiment, that variable is determined as a probability distribution of random variables. It can be discrete (not constant) or continuous or both.

Mostly, statisticians make use of capital letters to denote a probability distribution of random variables and small-case letters to represent any of its values.

X denotes the probability distribution of random variable X

P(X) denotes the probability of X. 

p( X=x) denotes the probability that random variable x is equivalent to any particular value, represented by X. For example: P (X=1) states the probability distribution of the random variable X is equivalent to 1. 

Probability Distribution Definition

Probability Distributions give up the possible outcome of any random event. It is also  identified on the grounds of underlying sample space as a set of possible outcomes of any  random experiment. These settings can be set of prime numbers, set of real numbers, set of complex numbers, or set of any entities. The Probability distribution is a part of probability and statistics.


Random experiments are termed as the outcomes of an experiment whose results cannot be predicted. For example- if we toss a coin, we cannot predict what will appear, either the head or tail. The possible result of a random experiment is known as the outcome. And the set of outcomes is termed as a sample point. Through these possibilities, we can design a probability table on the basis of variables and probabilities.

Types of Probability Distribution:

There are two types of probability distribution which are used for distinct purposes and various types of data generation processes.

These Two Types  of Probability Distribution are:

  1. Normal or Continuous probability Distribution

  2. Binomial or Discrete Probability Distribution

Normal Probability Distribution

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

For example- Set of real numbers,set of prime numbers, are the normal distribution examples as they provide all possible outcomes of real numbers and prime numbers. The real-life scenarios such as the temperature of a day is an example of continuous distribution.

As the normal distribution statistics predict some natural events clearly, it has developed a standard of recommendation for many probability issues. Some example are:

  • Rolling of a dice

  • Tossing a  coin

  • Height of a newly born babies

  • Size of men or women's shoes.

  • Income distribution of a country between rich and poor

Binomial / Discrete Probability Distribution

The Binomial distribution is also termed as a discrete probability function where the set of outcomes are discrete in nature. For example: if a dice is rolled, then all its possible outcomes will be discrete in nature and it gives the  mass of outcome. It is also considered a probability mass function

In a  real-life scenario the concept of binomial distribution is used for :

  • To find out the number of men and women working in a college

  • To find the number of used and unused particles while manufacturing a product

  • To check the number of people watching the  particular channel by calculating the or  yes or no.

  • To take a survey of positive and negative feedback for some issues.

Probability Distribution Formulas

Here are some of the probability distribution formulas based on their types. 


The Formula for the Normal Distribution


P(x) = \[\frac{1}{\sqrt{2\pi\sigma^{2}}}\] . \[e^\frac{{(x-\mu)^{2}}}{2\sigma^{2}}\]


Here, 


μ = Mean Value

σ  =Standard Deviation

x= Normal random variable

If mean μ = 0 , and standard deviation =1 ,then this distribution is termed as normal distribution.


The Formula for the Binomial Distribution

P(x) = \[\frac{n!}{r!(n-r)!}\] . \[p^{r}\]\[(1-p)^{n-1}\]

P(x) = C (n,r).\[p^{r}\]\[(1-p)^{n-1}\]


Here,


n=Total number of events

r= Total number of successful events

p = successful on a single trial probability,

1-p =  Failure probability

nCr =  [n!r!(n-r)!]

Probability Distribution Function

The functions which are used to define the distribution of probability are termed as a  probability distribution function.These functions can be defined on the basis of their types. These probability distribution functions are also  used in respect of probability density function for any of the given random variables.


In Normal distribution, the function of a real-valued random variable X is the function derived by:

Fx(x) =P(X ≤ x)


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

For the closed interval (a →b) the cumulative probability function can be identified as:


P( a< X ≤ b) = Fx (b) -Fx(a)


If the cumulative probability function is expressed as integral of the probability density function fx, then,

\[F_x(x)\] = \[\int_{x}^{-\infty}f_{x}(t)dt\]

In terms of a random variable X= b, cumulative probability function can be defined as:


P(X=b) = \[F_{x}\](b) - \[\lim_{x\rightarrow b}f_{x}(t)\]


As we know, the binomial distribution is determined as the probability of mass or discrete random variable which yields exactly some values. This distribution is also termed as probability mass distribution and the function linked with it is known as probability mass function.

For example 

A random variable X and sample space S is termed as

X:S  →A

And A ∈ R, where R is termed as a discrete random variable

Then , probability mass function fx : A -[0,1] or X can be termed as:


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

Probability Distribution Table Introduction

The probability distribution table is designed in terms of a random variable and possible outcomes. For instance- random variable X is a real-valued function whose domain is considered as the sample space of a random experiment. The probability distribution of P(X) of a random variable X is the arrangement  of numbers.


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

Probability Distribution Table

X

X1

X2

X3

…………...

XN

P(X)

P1

P2

P3

………….

pn


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


Solved Example

Here are some probability Distribution examples which will help you to understand the concept thoroughly:


  1. What is the probability of getting 7 heads, if a coin is tossed for 12 times?

Solution:

Number of trials (n) =12

Number Of success (r) - 7

Probability of single trial (p)= ½ = 0.5

nCr  = [n! /r!] X (n-r)!

=12! /7! (12-7)!

= 12! / 7! 5!

= 95040120

= 792

pr = 0.5 = 0.0078125

To find (1-p) n-r, calculate (1-p) and (n-r)

(1-p) =1-0.5 = 0.5

n-r = 12-7= 5

(1-p) n-r = (0.5)7 = 0.03125

Now calculate

P (X=r) nCr. pr. (1-p) n-r

=792 x 0.0078125 x 0.03125

=0.193359375

Hence, the probability of getting 7 head is 0.19


2. The probability of man hits the target is ¼ . If he fires 9 times, then find the probability that he hits the target exactly 4 times.

Solution:

Total number of fires (n) =9

Total number of success hites=r=4

Probability of hitting the targets -p=¼

Probability of not hitting the targets =q=1-p=1-(¼)=¾

Calculating nC

9C4 9! / (4! 5!) (9 *8 *7*6*5!) (4*2*2*1 *5!) = 126

Probability of the person hits the target exactly 4 times

→ 9C4  (1/4) (3/4) (9-4)

= 126 * (1/256) *(243/1024)

= 0.1168

Fun Facts

  • New probability theory was pioneered by Gerolama Cardano, Pierre de Fermat And Blaise Pascal in the 16th century.

  • The gambler dispute which took place in 1654 gave rise to the formation of the Mathematical theory of probability by two famous French Mathematician Pierre de Fermat And Blaise Pascal

  • Girloma Cardano is known as the “Father of Probability”.

Quiz Time

  1. Which is not possible in a probability in the following types?

  1. p(x) = 0.5

  2. p(x) = -0.5

  3. p(x) = 1

  4. Σ x p(x) = 3


2. In a binomial probability distribution, if n is the number of trials and p is the number of success, then mean value is given by

  1. np

  2. n

  3. p

  4. np(1-p)


3. Which of the following is not a feature of normal distribution?

  1. The mean value is always 0

  2. The area under the curve is equivalent to 1

  3. The mean, median and mode are similar

  4. It is a symmetrical distribution


4. It is perfect to use binomial distribution for

  1. Large values of 'n'

  2. Small values of 'n'

  3. Fractional values of 'n'

  4. Any values of 'n’

FAQ (Frequently Asked Questions)

1. Explain the Prior Probability and Posterior Probability

Prior Probability 

According to Bayesian statistical conclusion, a prior probability distribution, also termed as prior, of an unforeseeable quantity is the probability distribution, asserting one’s belief about this unforeseeable quantity prior to any proof is taken into consideration. For example- the prior probability distribution exhibits the relative proportion of the voters who will vote for some politicians in an upcoming election. The unknown quantity may be a parameter of the design or a possible variable instead of an observable variable.


Posterior Probability

The posterior probability is the possibility an event will take place after all the data or information have been taken into consideration. It is nearly linked to the prior probability where an event will take place before any data or new evidence taken into consideration. It is primarily a modification of prior probability. 


Formula to Calculate the Posterior Probability is Given Below:

Posterior Probability - prior probability + New evidence

These two probabilities are commonly used in Bayesian hypothesis testing. There are old data that says that around 70% of the college students will complete their graduation degree within 4 years.This is considered as a prior probability. If  we  think that the figures which came out are lower, then we will start collecting more data. The data collected signifies that the actual figure is indeed closer to 50% which is considered as a posterior probability.

2. What is Negative Probability Distribution?

In probability theory and statistics, if in a  binomial probability  distribution, the number of successes in a series of independent and similar scattered Bernoulli trials prior to  an individual number of failures takes place, then it is identified as a negative binomial distribution. Here the number of figures is represented as r. For example: if we throw a dice and examine the occurrence of 1 as a failure and all non -1’s as successes. Now, if we throw a dice periodically until 1 comes the third time i.e. r = three failures, then the probability distribution of the number of non-1s that appears would be the negative binomial distribution.