# More on Pareto distribution

There are several types of Pareto distributions. The Pareto distribution described in this previous post is called the Pareto Type II distribution (also called Lomax distribution). There is also a Pareto Type I distribution. The difference between the two Pareto types is that the support of Pareto Type I is the interval $(\theta, \infty)$. In other words, the random variable takes on positive real numbers larger than some fixed positive parameter $\theta$. On the other hand, Pareto Type II distribution can take on any positive real number. However, the two types are mathematically related. Their density functions have the same shape (if they have the same parameters).

Pareto Type I Density Curve (Red) versus Pareto Type II Density Curve (Green)

In the above diagram, the red curve is the density function for the Pareto Type I distribution while the green curve is the density function for the Pareto Type II distribution. For both distributions, the shape parameter is $\alpha=2$ and the scale parameter is $\theta=5$.

Note that the green density curve is the result of shifting the red curve horizontally 5 units to the left. The following are the density functions.

Red Curve
$\displaystyle f(x)=\frac{2 (5)^2}{x^3}=\frac{50}{x^3} \ \ \ \ \ x>5$

Green Curve
$\displaystyle g(x)=\frac{2 (5)^2}{(x+5)^3}=\frac{50}{(x+5)^3} \ \ \ \ \ x>0$

From the diagram and the density functions, we see that the green density curve is the result of shifting the red curve to the left by 5. Otherwise, both curve have the same distributional shape.

Since Pareto Type II is a shifting of Pareto Type I, they share similar mathematical properties. For example, they are both heavy tailed distributions. Thus they are appropriate for modeling extreme losses (in insurance applications) or financial fiasco or financial ruin (in financial applications). Which one to use depends on where the starting point of the random variable is.

However, the mathematical calculations are different to a large degree since the Pareto Type II density function is the result of a horizontal shift.

We would like to introduce blog posts in two companion blogs that discuss the Pareto distribution in greater details. This blog post is from a companion blog called Topics in Actuarial Modeling. It describes Pareto Type I and Type II in greater details. This blog post from another companion blog called Actuarial Modeling Practice has a side-by-side comparison of Pareto Type I and Pareto Type II. It also discusses the two Pareto types from a calculations stand point. This blog post has practice problems on the two Pareto types.

Another important mathematical property of Pareto Type II is that it is the mixture of exponential distributions with gamma mixing weights (exponential-gamma mixture). This is also discussed in the highlighted blog posts.

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$\copyright$ 2017 – Dan Ma