Pareto Distribution

Pareto distribution is a continuous probability distribution named after the Italian economist Vilfredo Pareto, who introduced the concept in 1897 while studying the distribution of wealth.

It is widely known for modelling phenomena where a small proportion of occurrences account for the majority of the effect. The distribution is often linked to the Pareto Principle, also called the 80/20 rule, which states that roughly 80% of effects come from 20% of causes.

Some Examples of Pareto Distribution Includes:

• In many economies, the Pareto principle applies to wealth distribution. A small percentage of individuals control a large portion of the total wealth.

• The distribution of internet traffic follows a Pareto distribution. A small number of websites receive a majority of the traffic. For instance, websites like Google, Facebook, and YouTube account for the majority of web traffic, while the majority of smaller websites receive only a fraction of the total.

• The population sizes of cities often exhibit a Pareto distribution. A few large cities (like London, New York, or Tokyo) contain a significant portion of the total population, while most cities are relatively small.

Properties of Pareto Distribution

Key Properties of Pareto Distribution are discussed below:

Probability Density Function (PDF) of Pareto Distribution

The probability density function of a Pareto distribution is given by:

Where:

• α is the shape parameter (also called the "tail index").
• x_m is the scale parameter, representing the minimum possible value.
• x is the value of the random variable.

The PDF shows that for higher values of α\alphaα, the distribution decays faster, meaning that extreme values are less probable.

Cumulative Distribution Function (CDF) of Pareto Distribution

The cumulative distribution function is the probability that the variable X is less than or equal to a value xxx. It is expressed as:

This shows the proportion of values less than or equal to x.

Parameters of Pareto Distribution

• Shape Parameter (α): Controls the "thickness" of the tail. Higher values of α\alphaα indicate a distribution with fewer large values (thinner tails), while smaller values indicate a heavy-tailed distribution.

• Scale Parameter (x_m): Represents the minimum possible value of the distribution. All values must be greater than or equal to this minimum.

Mean and Variance of Pareto Distribution

• Mean (Expected Value): E, the mean is undefined (the distribution has no finite mean).
• Variance: , the variance is infinite, meaning the distribution has no finite variance.

Pareto Principle (80/20 Rule)

A common application of the Pareto distribution is in the Pareto principle, or the 80/20 rule, which states that roughly 80% of effects come from 20% of the causes. This is a consequence of the skewed nature of the Pareto distribution, where a small proportion of occurrences contributes disproportionately to the total outcome.

Solved Questions on Pareto Distribution

Que 1. Suppose a random variable X follows a Pareto distribution with shape parameter α = 3α and scale parameter x_m = 2. Find the mean of the distribution.

Solution:

The mean E(X) of a Pareto distribution is given by:

Substitute the given values α = 3 and x_m = 2:

Thus, the mean of the distribution is 3.

Que 2. Find the mean of a Pareto distribution with parameters α = 3 and x_m = 2.

Solution:

Use the mean formula: Mean
Mean =

Thus, Mean = 3

Que 3. Suppose X follows a Pareto distribution with shape parameter α = 2 and scale parameter x_m = 1. What is the probability that X ≥ 3?

Solution:

The cumulative distribution function (CDF) of the Pareto distribution is:

We are looking for P(X ≥ 3). This is given by:
P(X ≥ 3) = 1 − P(X < 3) = 1 − F(3)
F(3) = 1 − (1/3)² = 1 − 1/9 = 8/9

So, P(X ≥ 3) = 1 − 8/9 = 1/9​

Thus, the probability that X ≥ 3, or approximately 0.1111.

Practice Question on Pareto Distribution

Q 1. A random variable X follows a Pareto distribution with shape parameter α = 4 and scale parameter x_m = 3. Find the mean of the distribution.

Q 2. If X follows a Pareto distribution with shape parameter α = 2.5 and scale parameter x_m = 1, what is the probability that X ≥ 5?

Q 3. Let X follow a Pareto distribution with α = 3.5 and x_m = 7. Find the variance of the distribution.

Q 4. If X follows a Pareto distribution with shape parameter α = 5 and scale parameter x_m = 2, find the probability density function (PDF) of the distribution.

Q 5. Consider a country where the income follows a Pareto distribution with shape parameter α = 1.7 and scale parameter x_m = 1200. What proportion of the population holds 90% of the wealth?

Answer Key

1. Mean: 4
2. Probability X ≥ 5: 0.0179
3. Variance: 18.3
4. PDF: f(x)=160/x⁶​ for x ≥ 2
5. Proportion of population holding 90% of the wealth: 10%

Conclusion

Pareto distribution helps explain how in many areas of life, a small number of things have a large impact. Whether it's wealth, internet traffic, or natural disasters, a few big events or individuals often dominate the outcomes.

Read More,

• Probability in Maths
• Random Variable
• Discrete Random Variable
• Continuous Random Variable
• Probability Distribution Function