This applet computes probabilities and percentiles for gamma random variables:
$$X \sim Gamma(\alpha, \beta)$$
When using rate parameterization, replace $\beta$ with $\frac{1}{\lambda}$ in the following equations.
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
$$f(x)=\frac{1}{\Gamma(\alpha)\beta^\alpha} x^{\alpha-1} e^{-x/\beta}$$
where $x > 0$, $\alpha > 0$, and $\beta > 0$
- $\mu=E(X)=\alpha\beta$
- $\sigma^2=Var(X)=\alpha\beta^2$
- $\sigma=SD(X)=\sqrt{\alpha\beta^2}$