Pareto Distribution Mode Calculator

Pareto Distribution Mode Calculator

Enter the scale parameter \( x_m \) and the shape parameter \(\alpha\) to calculate the mode of the Pareto distribution.

* Note: The mode of the Pareto distribution is always equal to the scale parameter \( x_m \), regardless of \(\alpha\).

Step 1: Enter Parameters

e.g., 1

e.g., 2 (Note: This value does not affect the mode)

How It Works

The PDF of the Pareto distribution is given by: $$ f(x) = \frac{\alpha x_m^\alpha}{x^{\alpha+1}}, \quad x \ge x_m. $$

Since the PDF is a strictly decreasing function for \( x \ge x_m \), the maximum value (mode) occurs at \( x = x_m \). Thus, the mode is: $$ \text{Mode} = x_m. $$

Formula: \( \text{Mode} = x_m \)

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