Common Probability Distributions | Definition and Types of Probability Distribution

Probability Distribution is a statistical function using which the probability of occurrence of different values within a given range can be calculated. It is a function that gives the relative likelihood of occurrence of all possible outcomes of an experiment.

Let’s consider a random event of throwing dice, it can return 6 possible values (1,2,3,4,5,6). The probability distribution that 1 will be returned is ~17%.

Probability Density Function (PDF)

The Probability Density Function represents the density of a continuous random variable lying between a specific range of values.

Cumulative Density Function (CDF)

The cumulative Density Function calculated for a random variable R at a point x represents the probability distribution that R will have a value less than or equal to x.

Probability Mass Function (PMF)

The Probability Mass Function represents the probability distribution of a discrete random variable.

Types of Probability Distributions

  1. Discrete Distribution
  2. Continuous Distribution
Discrete variables represent  data that is countable (eg. number of apples).
Continuous data is data that falls in a constant sequence (eg. temperature, 

Discrete Distribution

Discrete Distribution represents the probability distribution of discrete data. A few discrete distributions are:

  1. Binomial Distribution
  2. Bernoulli Distribution
  3. Uniform Distribution
  4. Poisson Distribution

1. Binomial Distribution

The Binomial Distribution is used when there are only two possible outcomes, success (1) or failure (0) for n number of trials. The probability for both outcomes is the same in all the trials.


  1. Each trial is independent.
  2. There are only two possible outcomes 1 and 0.
  3. The probability of success and failure is the same for all the trials.

Probability Mass Function


  • P(x) = Binomial Probability
  • n = number of trials
  • p = probability of success
  • q = probability of failure
  • x = number of times for a specific outcome within n trial.


2. Bernoulli Distribution

Similar to the Binomial  Distribution, Bernoulli Distribution has only two possible outcomes, success (1) or failure (0) but only one trial.


  1. The number of trails is 1.
  2. There are only two possible outcomes 1 and 0.
  3. The probability of success and failure may not be the same.

Bernoulli Distribution is similar to the Binomial Distribution. The only difference is that in Bernoulli, n=1 always, and x will take a value of 0 or 1.

Probability Mass Function

 where x ∈ (0,1)

You can derive the probability mass function of the Bernoulli distribution from the pmf of the Binomial distribution by keeping n=1 and x =0 or 1 in the equation.

3. Uniform Distribution / Rectangular Distribution

When you roll a die, the probability of getting each value (from 1 to 6) is equal (1/6). This is a perfect example of Uniform Distribution.


  • Uniform distribution is a probability distribution where all outcomes are equally likely.
  • This is a distribution that has constant probability.

Density Function

 for -∞ < a ≤ x ≤ b < ∞
  • Area of Rectangle = length * width.
  • Area of Rectangle = (b-a)*(1/(b-a)) =1.
  • the area under the curve is always 1.

4. Poisson Distribution

Poisson Distribution can be used to find the probability of several events in a time period. For example, imagine you have a clinic and want to find out approximately how many patients visit the clinic in a day. It can be any number. Now the total number of patients visited in a day is calculated using Poisson Distribution.


  1. All events occur independently.
  2. An event can occur any number of times.

Probability Mass Function

 where u is the mean


Continuous Distribution

Continuous Distribution represents the probability distribution of continuous data. A few continuous distributions are:

  1. Normal Distribution
  2. Standard Normal Distribution
  3. Student’s T Distribution
  4. Chi-Squared Distribution

1. Normal Distribution / Gaussian Distribution

A Random Variable (X) having mean (\mu) and standard deviation (\sigma) is said to be Normally Distributed if it has the below properties:


  • Mean = Median = Mode
  • No Skewness
  • It follows Bell Curve
  • It is symmetrical on both sides of the mean

Empirical Rule

  • If you go one standard deviation to the left and one standard deviation to the right, it covers 68% of the total data.
  • If you go two standard deviations to the left and two standard deviations to the right, it covers 95% of the total data.
  • If you go three standard deviations to the left and three standard deviations to the right, it covers 99.7% of the total data.
Normal Distribution
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Probability Density Function

probability density function
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2. Standard Normal Distribution

Standard Normal Distribution is a special type of Normal Distribution where the mean is 0 and the standard deviation is 1.

How to convert Normal Distribution into Standard Normal Distribution?

Normal Distribution can be converted to Standard Normal Distribution by using a z-score. This process is also called Standardization.

z-score = 


  •   = mean
  • = standard deviation
  • = current value

3. Student’s T Distribution

When the sample size of the data is small, instead of following Normal Distribution it follows Student’s T Distribution.

Student’s T-Distribution is used when the sample size is low and the population standard deviation is unknown.

Let n be the sample size, then the Degree of Freedom is n-1. As the degree of freedom increases, the student’s t distribution becomes a normal distribution.

4. Chi-Squared Distribution

The Chi-Squared distribution is used to describe the distribution of a sum of squared random variables.

The mean of the chi-square distribution is equal to the degrees of freedom (k) while the variance is twice the degrees of freedom (2k).

The shape of the chi-square distribution depends on the number of degrees of freedom k. When k is small, the shape of the curve tends to be skewed to the right, and as the k gets larger, the shape becomes more symmetrical and approaches a normal distribution.


  • It is used to test the goodness of fit of a distribution of data.
  • The value of the chi-squared distribution ranges from 0 to ∞.

End Notes

Thank you for reading this article. By the end of this article, we are familiar with different Probability distributions that are frequently used in Statistics.

I hope this article was informative. Feel free to ask any query or give your feedback in the comment box below.