For any Data Analytics/ Scientist, a professional or a student, probability distribution is an unquestionable requirement to know. It gives the principle foundation to analytics and inferential statistics. While the theory of probability covers the numerical and mathematical aspects, distributions assist us visualizing what's going on underneath.
In this article, some important probability distributions have been discussed.
Types of Common Data
6 Important Distribution Types
1. Bernoulli Distribution
It is the easiest distribution to understand with only two possible outcomes
Success (1)
Failure (0)
Example:  While tossing a coin just once you will either get a Head or a Tail. So the random variable X having a Bernoulli distribution can take value 1 with the probability of success,p, where p=0.5, and the value 0 with the probability of failure, q or 1p, where q= 0.5.
The probabilities of success and failure may or may not be equally likely.
 Probability Mass Function (p.m.f)
px(1p) x
 Mean/ Expectation of random variable X having Bernoulli Distribution
μ= E(X)= 1*p + 0*(1p)
 Variance of a random variable X having Bernoulli Distribution
V(X)= E(X2)  [E(X)]2= pp2= p(1p)
 Examples: 
 A new born baby is either a girl or a boy
 You either pass or fail in an examination
 It will either rain or not rain tomorrow
2. Binomial Distribution
Let's take our favourite sport into consideration, cricket, here. If you win a toss today that doesn't mean that you will win the toss tomorrow too. It is a case where Binomial Distribution is considered. Properties
 Each trial is independent of each other
 There are only two possible outcomes in a trial which is either a success or a failure
 Total number of 'n' trials are carried out
 As all trials are identical, the probability of success and failure is same for all of them.
 Probability Mass Function (p.m.f)
 Mean/ Expectation of Binomial Distribution
μ = E(X)= n*p
 Variance of Binomial Distribution
V(X) = n*p*q
 Bernoulli Distribution is considered to be a special case of Binomial Distribution with the difference that the former has a single trial
3. Uniform Distribution
The basis of Uniform Distribution is that, unlike Bernoulli Distribution, the probabilities of 'n' possible outcomes are equally likely. It is also called a rectangular distribution, which has constant probability.
It characterizes a condition where all outcomes in a range between a minimum and maximum value are equally probable. One of the many examples could be the number of bouquets sold everyday at a flower store is uniformly distributed with a maximum of 40 and a minimum of 10.
 Probability Density Function (p.d.f) of variable X having Uniform Distribution is
'a' & 'b' are the parameters of Uniform Distribution
 Mean/ Expectation of Uniform Distribution
μ= E(X) = (a+b)/2
 Variance of Uniform Distribution
(ba)2 / 12
 Standard Uniform Probability Density Function
Where a= 0 & b= 1
4. Normal Distribution
A normal distribution is an organization of a data set in which maximum values congregate in the centre of the range and the rest taper off symmetrically toward either ends. Properties
 Mean= Median= Mode i.e. symmetry about the centre (50% of values lower than the mean & 50% more than the mean
 It has a bell shaped curve
 Area under the bell shaped normal curve is 1.
 Two parameters define Normal Distributions Mean (μ)& Standard Deviation (σ)
 68% of the values of a normal distribution is within one std. dev. of the mean.
 95% of the values of a normal distribution is approximately within two std. dev. of the mean.

Probability Density Function (p.d.f) of variable X having Normal Distribution is
 Mean/ Expectation of Normal Distribution
μ = E(X)
 Variance of Norma; Distribution
V(X) = σ^{2}
 Examples: 
 Marks in a Test
 Errors in Measurement
 Heights of People
 Blood Pressures
 Size of products produced by machines
5. Poisson Distribution
It is useful for describing events with extremely low probabilities of occurrence within fixed interval of time or space. Properties
 The outcome of a successful event should not be influenced by any other successful event
 Probability of success within a short period of time = Probability of success within a longer period of time
 Probability of success > 0 as the interval gets smaller
 Probability Mass Function (p.m.f)
 Mean/ Expectation of Poisson Distribution
μ = E(X) = λt; t= time interval in which the event occurs
 Variance of Poisson Distribution
V(X) = μ = λt
λ = Rate at which an event occurs
t = length of a time interval
X= Number of events that occur in that time interval
 Examples: 
 Number of crimes that are reported in a place on a day
 Number of calls of emergency that are reported at a hospital in a day
 Number of customers for car services in an hour
 Number of typing errors on every page of a newspaper
6. Exponential Distribution
How long will you have to wait before a patient gets in your clinic? How much time will elapse before an earthquake hits a given region? How long will it take before a call centre receives the next call? How long will an equipment function before it seizes up?
Problems like these are generally solved in probabilistic terms using Exponential Distribution.
Exponential distribution characterizes the interval of time between the events. Exponential distribution is extensively used for survival analysis. From the expected life of a machine to the expected life of a human, exponential distribution successfully provides the result.
 Probability Density Function (p.d.f) of a random variable X having Exponential Distribution
Parameter λ> 0 is called the rate
 Mean of X having Exponential Distribution
E(X)= 1/λ
 Variance of X having Exponential Distribution
V(X)= 1/λ2

Examples: 
 Life of a Refrigerator
 Length of time between arrivals at a hospital
 Length of time between train arrivals
Probability Distributions are pervasive in numerous segments, in particular, finance & insurance, physics, engineering, computer science and even social sciences in which the students of psychology and healthcare are broadly utilizing probability distribution. It has a simple & extensive application.
#ReadyBusinessPlan #Ask3EA #LearnAt3EA #3EA
#BusinessPlan #CapacityEnhancement #CapacityBuilding #Capacity #Assessment
#Global #DataAnalyst #DataScience #DataAnalysis