All about Bernoulli, Binomial and Poisson Distribution

#statistics #distribution #machine_learning

Ankit Narang Sept 27 2020 · 2 min read
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Random Variables

A random variable is a quantity produced by random process.

Mainly there are two kinds of random variables in practice. The first one is discrete random variable and the other is continuous random variable.

Discrete random variable is the one which are produced by counting values, such as the integers 0,1,2,3… and so on. For example number of products you buy would be a discrete random variable.

Continuous random variable on the other hand is the data which is obtained by taking measurements. Heights and weights are the two popular examples of continuous random variable.

Now let us move on to the distributions and understand how they are different from each other.

Bernoulli Distribution

Bernoulli distribution is a discrete probability distribution that covers a case where an event will have a binary outcome such as either 0 or 1. One very common example would be a flip of two sided coin. When you flip a coin you get either heads or tails. Hence, we can say that this kind of experiments follow a Bernoulli distribution.

As in the Bernoulli distribution, we only have two outcomes namely 1 (success) and 0 (Failure) So the random variable can take the probability of p when the outcome turns out to be a success or we can obtain the probability of the other outcome as q which can be calculated by (1-p) as there are only two events.

Use case in machine learning? We can see various use case of Bernoulli random variable in machine learning as well when we have only two outcomes. Netflix uses a variable which tells whether a user likes a movie or not.

Binomial Distribution

The outcomes when multiple Bernoulli trials are performed represents the binomial distribution.

The main relation between Bernoulli distribution and binomial distribution arises with the number of times a trial is performed. The binomial distribution tries to summarize the number of successes k in a given number of Bernoulli trials n, with a probability of success for each trial.

Some example uses include the number of heads in n coin flips, the number of disk drives that crashed in a cluster of 1000 computers, and the number of advertisements that are clicked when 40,000 are served.

The formula for computing probability of binomial distribution variables is as follows:

 Poisson Distribution

The distribution where time & space are fixed but number of count is not fixed is known as Poisson distribution. Examples of variable which follows the Poisson distribution are given below:

1.      Number of customers arriving in every one minute interval.

2.      Surface defects on a new refrigerator.

 There are two important concepts of Poisson distribution namely event of interest and area of opportunity. In the first example above we can say that the event of interest is customer arriving and area of interest would be one minute interval. We can conclude that whenever time interval is specified the variable will be from a Poisson distribution given that number of count is fixed.

The formula to compute the probability of Poisson distribution variables is given below:

   X = Number of events (1,2,3..)   λ = expected number of units

So, this was a very gentle introduction to three of the very important distributions in statistics.  This will help you to make a connection of the variables in your dataset with various kinds of distributions in statistics and make the statistical analyses easy for you.

References:

https://web.stanford.edu/class/archive/cs/cs109/cs109.1178/lectureHandouts/070-bernoulli-binomial.pdf

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