### Pareto (Type I) Distribution$X \sim Pareto(x_m, \alpha)$

 $x_m=$ $\alpha=$ $x=$ P(X > x) = P(X < x) =

This applet computes probabilities and percentiles for Pareto (Type I) random variables: $$X \sim Pareto(x_m, \alpha)$$

#### Directions

• Enter the scale $x_m$ and the shape $\alpha$.
• To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue box and press "Enter" or "Tab" on your keyboard. The probability $P(X \lt x)$ will appear in the pink box. Select $P(X \gt x)$ from the drop-down box for a right-tail probability.
• To determine a percentile, enter the percentile (e.g. use 0.8 for the 80th percentile) in the pink box and select $P(X \lt x)$ from the drop-down box. The percentile $x$ will appear in the blue box.

On the graph, the $x$ value appears in blue while the probability is shaded in pink.

#### Details

• Probability density function $$f(x)=\frac{\alpha x_m^\alpha}{x^{\alpha+1}}$$ where $x > x_m$, scale $x_m > 0$, and shape $\alpha > 0$
• $\mu=E(X)=\frac{\alpha x_m}{\alpha-1}$ for $\alpha > 1$ ($\infty$ otherwise)
• $\sigma^2=Var(X)= \frac{\alpha x_m^2}{(\alpha-1)^2(\alpha-2)}$ for $\alpha > 2$ ($\infty$ otherwise)
• $\sigma=SD(X)=\sqrt{\frac{\alpha x_m^2}{(\alpha-1)^2(\alpha-2)}}$ for $\alpha > 2$ ($\infty$ otherwise)