This applet computes probabilities and percentiles for beta random variables:
$$X \sim Beta(\alpha, \beta)$$
Directions
- Enter the shape $\alpha$ and the shape $\beta$.
- 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 and press "Tab" or "Enter" on your keyboard.
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
$$f(x)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1}$$
where $0 \le x \le 1$, $\alpha > 0$, and $\beta > 0$
- $\mu=E(X)=\frac{\alpha}{\alpha+\beta}$
- $\sigma^2=Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$
- $\sigma=SD(X)=\sqrt{\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}}$