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In the realm of Probability and Statistics, the Beta Distribution offers a continuous distribution defined by two positive parameters, \( \alpha \) and \( \beta \). These parameters shape the distribution, allowing it to model a variety of different types of data. The Probability Density Function (PDF) of a Beta distribution provides the probability of a random variable falling within a particular range. Given \( \alpha > 0 \) and \( \beta > 0 \), the PDF is typically expressed as follows:
\[ f(x; \alpha, \beta) = \frac{x^{\alpha - 1}(1-x)^{\beta - 1}}{B(\alpha, \beta)} \quad \text{for} \quad 0 < x < 1.\]
The term \( B(\alpha, \beta) \) is known as the Beta function, which is essentially the normalization constant that ensures the total probability integrates to one. This PDF enables us to describe the likelihood of different outcomes, making it a powerful tool in areas like Bayesian statistics and in scenarios involving proportions or probabilities.
Understanding the shape and behavior of the PDF can guide us to interpret real-world data more appropriately, as it gives insight into the distribution of values across the interval \([0,1]\).

The mean, mode, and variance are fundamental characteristics of any probability distribution and help summarize the data's behavior. For the Beta Distribution:

• Mean (\( \mu \)): This average value can be calculated using the formula, \[ \mu = \frac{\alpha}{\alpha + \beta}. \] It provides a measure of central tendency, representing the average outcome of an experiment.
• Mode: The mode indicates the most likely outcome or the peak of the distribution. It is computed when both \( \alpha \) and \( \beta \) are greater than 1 using \[ \text{Mode} = \frac{\alpha - 1}{\alpha + \beta - 2}. \]
• Variance (\( \sigma^2 \)): Variance assesses the spread or "dispersion" of data around the mean and is essential for understanding variability. For the Beta Distribution, it is given by, \[ \sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}. \]

The mean informs about the center, the mode tells the most frequent value, and the variance provides insight into how much the values spread out. Together, these metrics describe the shape and spread of the Beta distribution.

Calculating the parameters of a Beta Distribution such as the mean, mode, and variance requires substituting the values of \( \alpha \) and \( \beta \) into their respective formulas. These involve straightforward substitution but are essential for characterizing the distribution.
Let's consider the values provided:

• For \( \alpha = 3 \) and \( \beta = 1.4 \), plugging into the formulas yields:
  • Mean: \( \mu = \frac{3}{4.4} \approx 0.6818 \)
  • Mode: \( \frac{2}{2.4} \approx 0.8333 \)
  • Variance: \( \sigma^2 \approx 0.0360 \)
• For \( \alpha = 10 \) and \( \beta = 6.25 \):
  • Mean: \( \mu = \frac{10}{16.25} \approx 0.6154 \)
  • Mode: \( \frac{9}{14.25} \approx 0.6316 \)
  • Variance: \( \sigma^2 \approx 0.0134 \)

These parameter calculations illustrate the different characteristics of the Beta Distributions under varying conditions. While the mean slightly changes, the variance shows more pronounced differences, indicating different levels of spread or concentration around the mean. These illustrative examples show how the beta parameters influence distribution properties.