### Weibull Distribution$X \sim Weibull(\alpha, \beta)$

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

This applet computes probabilities and percentiles for Weibull random variables: $$X \sim Weibull(\alpha, \beta)$$

#### Directions

• Enter the shape $\alpha$ and the scale $\beta$.
• 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 $$f(x)=\frac{\alpha}{\beta} \left( \frac{x}{\beta} \right)^{\alpha-1} e^{-(x/\beta)^\alpha}$$ where $x > 0$, shape $\alpha > 0$, and scale $\beta > 0$
• $\mu=E(X)=\beta\,\Gamma\left(1+\frac{1}{\alpha}\right)$
• $\sigma^2=Var(X)=\beta^2 \left( \Gamma\left( 1 + \frac{2}{\alpha} \right) - \Gamma\left( 1 + \frac{1}{\alpha} \right)^2 \right)$
• $\sigma=SD(X)=\sqrt{\beta^2 \left( \Gamma\left( 1 + \frac{2}{\alpha} \right) - \Gamma\left( 1 + \frac{1}{\alpha} \right)^2 \right)}$