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

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

This applet computes probabilities and percentiles for gamma random variables: $$X \sim Gamma(\alpha, \beta)$$ When using scale parameterization, let $\beta = \frac{1}{\theta}$ in the following equations.

#### Directions

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