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)