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

- Discrete Distribution
- Continuous Distribution

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

## Discrete Distribution

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

- Binomial Distribution
- Bernoulli Distribution
- Uniform Distribution
- 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.

#### Properties

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

#### Probability Mass Function

where

- 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.

#### Example

## 2. Bernoulli Distribution

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

#### Properties

- The number of trails is 1.
- There are only two possible outcomes 1 and 0.
- 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

wherex∈ (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.

#### Properties

- 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.

#### Properties

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

#### Probability Mass Function

whereuis the mean

#### Example

## Continuous Distribution

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

- Normal Distribution
- Standard Normal Distribution
- Student’s T Distribution
- Chi-Squared Distribution

## 1. Normal Distribution / Gaussian Distribution

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

#### 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.

#### Probability Density Function

## 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 =

where

- = 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.

#### Properties

- 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.