A Gentle introduction to Probability Distributions – Part 2

This is Second Part of  “A Gentle introduction to Probability Distribution”  for part one click here.

Discrete Probability Distribution

In your Statistics class you must have encountered distributions like Bernoulli , Binomial, etc. All these distributions comes under the heading of Discrete Probability Distribution. if the probability of a random variable comes out to be discrete or single valued rather than a range.

In this article we will discuss definitions and mathematical formulation of some of the most popular discrete distributions :

  1. Bernoulli distribution
  2. Binomial distribution
  3. Poisson distribution

Bernoulli distribution

In an experiment whose outcome can be classified as either a success or failure. Let ‘X=1’ if outcome is success and ‘X=0’ if outcome is failure , then the probability mass function of X is given by

P(0) = P(X=0) = (1-P)
P(1) = P(X=1) = P,

Where P is Probability of success , then we call X is Bernoulli random variable.

Properties of Bernoulli random variable

Mean : P
Variance : P(1-P)
skewness : \frac{1-2p}{\sqrt{p(1-p)}}
Kurtosis :\frac{1-6p(1-p)}{p(1-p)}

 

Binomial Distribution

Let’s suppose ‘n’ independent experiments conducted , each of which results in a success with probability ‘P’ and failure with (1-P) .If ‘X’ represents the number of success in all ‘n’ experiments then X is called Binomial random variable with parameters ‘n’,’p’ where ‘n’ is number of trials and ‘p’ is probability of success in each trial.Let’s understand PMF of Binomial graphically first,

Source : Wikipedia

The probability mass function of Binomial distribution having parameter ‘n’,’p’ is given by,

p(k) = \Sigma_{k=0}^{n} ( p^{k} * (1-p)^{n-k}

Properties of Binomial distribution

Mean : n*p
Variance : n*p(1-p)
Skewness : \frac{1-2p}{\sqrt{np(1-p)}}
Kurtosis : \frac{1-6p(1-p)}{np(1-p)}

 

Poisson Distribution

This Discrete distribution describes the probability of a given number of events occurring in a given time interval, if these events occur with a constant rate and independent of each other.The Random variable X , taking on one of the values 0,1,2 is said to Poisson Random variable with parameters \lambda > 0 .Let’s take a look at PMF of Poisson random variable graphically .

if these events occur with a known constant rate and independently of the time since the last event. Source : wikipedia

The Pmf of Poisson distribution is given by,

p(i) => p(x=i) = \frac{e^{-\lambda}.\lambda^{i}}{{p!}}

Properties of Poisson Distribution

Mean : \lambda
Variance : \lambda
Skewness : \lambda^{-1/2}
kurtosis :  \lambda^{-1}

Probability distributions are very power tools in understanding data and its properties as if the data follows certain distribution then properties of that distribution are also applicable on that data. We will be covering many such interesting Machine Learning  topics in future so stay connected.

Happy Machine Learning”

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