https://file.aiquickdraw.com/mathbot/file/17277258960244nnc8xqa.jpg

Make use of the above result to find the mean, variance, and mode of the beta distribution.

The integral in the image represents the Beta function \( B(a, b) \), which is closely related to the Gamma function:

\[
B(a, b) = \int_0^1 \mu^{a-1} (1 - \mu)^{b-1} d\mu = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a + b)}
\]

### Beta Distribution
The Beta distribution, denoted as \( \text{Beta}(a, b) \), has the following probability density function (PDF):

\[
f(\mu; a, b) = \frac{\mu^{a-1}(1 - \mu)^{b-1}}{B(a, b)}
\]

Where \( \mu \in [0, 1] \), and \( a, b > 0 \) are shape parameters. The Beta function \( B(a, b) \) normalizes the distribution.

### Mean of the Beta Distribution

The mean of the Beta distribution is given by:

\[
\mathbb{E}[\mu] = \frac{a}{a + b}
\]

### Variance of the Beta Distribution

The variance of the Beta distribution is given by:

\[
\text{Var}[\mu] = \frac{ab}{(a + b)^2 (a + b + 1)}
\]

### Mode of the Beta Distribution

For \( a, b > 1 \), the mode of the Beta distribution is the value of \( \mu \) that maximizes the PDF. The mode is given by:

\[
\text{Mode} = \frac{a - 1}{a + b - 2}
\]

Thus, using the Beta function result provided, the mean, variance, and mode of the Beta distribution are:

- **Mean:** \( \frac{a}{a + b} \)
- **Variance:** \( \frac{ab}{(a + b)^2 (a + b + 1)} \)
- **Mode:** \( \frac{a - 1}{a + b - 2} \) (valid for \( a, b > 1 \)).

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Let me know if you would like further details or explanations.

Here are 5 related questions:
1. How does the Beta distribution relate to other common distributions like the uniform or binomial?
2. What is the Beta function and how is it connected to the Gamma function?
3. Can the Beta distribution represent probabilities over an interval other than [0, 1]?
4. How does the variance change as \( a \) and \( b \) increase or decrease?
5. What is the impact of \( a = 1 \) and \( b = 1 \) on the Beta distribution's shape?

**Tip:** The Beta distribution is often used as a prior in Bayesian statistics due to its flexibility in representing different types of beliefs.

Learn how to compute the mean, variance, and mode of the Beta distribution using the Beta function. This detailed explanation covers key mathematical concepts including the Beta and Gamma functions and is suitable for undergraduate students.

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