The Beta distribution is a continuous probability distribution defined on the interval [0, 1], and it is particularly useful in modeling random variables that represent proportions or probabilities. It is characterized by two positive shape parameters, α (alpha) and β (beta), and is defined by the following probability density function (PDF):
\[f(x; \alpha, \beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\,\Gamma(\beta)} \; x^{\alpha-1}(1-x)^{\beta-1}\]Where:
- $\alpha$: Shape parameter (alpha)
- $\beta$: Shape parameter (beta)
The key summary statistics for the Beta distribution are given by:
Mean:
\[\text{Mean} = \frac{\alpha}{\alpha + \beta}\]Variance:
\[\text{Variance} = \frac{\alpha \, \beta}{(\alpha+\beta)^2 (\alpha+\beta+1)}\]Mode:
\[\text{Mode} = \begin{cases} \frac{\alpha - 1}{\alpha + \beta - 2}, & \text{if } \alpha, \beta > 1 \\ \text{undefined}, & \text{otherwise} \end{cases}\]👉 Beta Distribution Visualization
![Graph of a Beta Distribution with α=5.1 and β=2, showing the probability density function over the interval [0, 1]](192252.png)
Key Features and How to Use:
Adjust α (Alpha):
Modify the alpha parameter to see how it affects the shape of the distribution. Higher values of α shift the distribution towards 1.
Adjust β (Beta):
Modify the beta parameter to observe its impact. Higher values of β shift the distribution towards 0.
Visualize the Probability Density:
The plot dynamically updates to display the Beta PDF for the selected parameters, illustrating how the density function behaves over the interval [0, 1].
Examine Summary Statistics:
The app calculates and displays key summary statistics such as the mean, variance, and mode (when defined) of the distribution, offering further insights into its properties.