Types of Probability Distribution
What is probability distribution
Probability distribution can be defined as a function which describes the likelihood of obtaining the possible values that a random variable can assume.
Types of Probability Distributions
Below are the types of probability distribution and I’ll explain few of them in details later in this article.
- Bernoulli Distribution
- Uniform Distribution
- Binomial Distribution
- Normal Distribution
- Poisson Distribution
- Exponential Distribution
1. Bernoulli Distribution
A Bernoulli distribution has only two possible outcome and a single trial. Outcomes could be either 1(success) or 0 (failure). So, a random variable X can take the value 1 with probability P (let’s say) and 0 with the probability q or 1-p.
Let us understand it with an example, we tossed a coin the possible outcomes would be either head or tail and we are taking occurrence of head as success and occurrence of tail as failure.
So, probability of getting head = probability of tail = 0.5.
For a mass function it can be concluded as,
P(x) = 1-p, x=0
= p, x= 1
Or, px(1-p)1-x where x € (0, 1).
Here one more thing need to taken care, as the probability of success and failure need not to be equally likely.
2. Binomial Distribution
Binomial distribution is also called a discrete probability distribution in which each trial is independent with two possible outcomes either success or failure. In this distribution the number of observations or trials is fixed. In normal distribution the mean, median and mode coincide and the area under the distribution curves would be 1 with bell shaped and symmetric about the line x=μ.
The formula for the binomial distribution is;
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
3. Normal Distribution
In normal distribution, the set of possible outcomes can take on values on a continuous range that’s why this is also known as the continuation/ cumulative distribution.
Formula for the normal distribution,
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
Few examples for the normal distribution are given as,
1. Tossing a coin.
2. Weight of population of India.
3. Height of new-born baby girls.
- Poisson Distribution
This is applicable in situations where events occur at random points of time and space and we are intended to know only in the number of occurrences of the event.
Conditions for the Poisson random variable:
- The number of successes in two separate time intervals is independent.
- The probability of a success during a small-time interval is proportional to the entire length of the time interval.
Few examples of Poisson distribution are:
1. Number of patients arriving in emergency ward every hour.
2. Number of tourists arrived in a particular city.
The PMF of X following a Poisson distribution is given by:
The mean µ is the parameter of this distribution. µ is also defined as the λ times length of that interval.
References
2. https://www.analyticsvidhya.com/blog/2017/09/6-probability-distributions-data-science/
3. https://byjus.com/maths/probability-distribution/
4. https://www.intmath.com/counting-probability/13-poisson-probability-distribution.php