# 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,
age).

## 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.

#### Properties

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

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.

## 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

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.

#### 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

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 () 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.

## 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 ∞.