Problem 232 - Applied Statistics and Probability for Engineers

Question

Determine the mean and variance of a beta random variable. Use the result that the probability density function integrates to \(1 .\) That is, \(\frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha+\beta)}=\int_{0}^{1} x^{\alpha-1}(1-x)^{\beta-1}\) for \(\alpha>0, \beta>0\)

Step-by-Step Solution

Answer

Mean: \( \frac{\alpha}{\alpha + \beta} \), Variance: \( \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \)."

Step 1: Recall Definitions

The probability density function (pdf) of a Beta distribution with parameters \( \alpha \) and \( \beta \) is given by \( f(x) = \frac{ ext{B}(\alpha, \beta)}{1} x^{\alpha-1} (1-x)^{\beta-1} \) for \( 0 < x < 1 \). The Beta function \( \text{B}(\alpha, \beta) \) is \( \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} \).

Step 2: Calculate Expected Value

The expected value (mean) \( E(X) \) of a Beta distribution is calculated using the formula \( E(X) = \int_{0}^{1} x f(x) \, dx \). For the Beta distribution, this simplifies to \( E(X) = \frac{\alpha}{\alpha + \beta} \).

Step 3: Compute Expected Value of Square

To find the expected value of \( X^2 \), use the formula \( E(X^2) = \int_{0}^{1} x^2 f(x) \, dx \). This results in \( E(X^2) = \frac{\alpha(\alpha + 1)}{(\alpha + \beta)(\alpha + \beta + 1)} \).

Step 4: Determine Variance

The variance \( \text{Var}(X) \) is defined as \( \text{Var}(X) = E(X^2) - [E(X)]^2 \). Substituting the values found in Steps 2 and 3, \( \text{Var}(X) = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)} \).

Step 5: Summary

The mean of the Beta distribution is \( \frac{\alpha}{\alpha + \beta} \) and the variance is \( \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \).

Key Concepts

Probability Density Function

The Probability Density Function (PDF) is a crucial concept in understanding continuous probability distributions. For the Beta distribution, this function helps in modeling a wide range of phenomena, such as the likelihood of different outcomes between 0 and 1.
The PDF of a Beta distribution is given by:

• \( f(x) = \frac{\text{B}(\alpha, \beta)}{1} x^{\alpha-1} (1-x)^{\beta-1} \)

where \(\text{B}(\alpha, \beta)\) is the Beta function, serving as a normalization constant. The parameters \(\alpha\) and \(\beta\) shape the distribution.
To maintain the essential property of a PDF, the area under the curve must equal 1. This condition ensures that the total probability of all possible outcomes is 1, making distributions like Beta both reliable and meaningful for probability calculations.

Expected Value

Expected Value, often thought of as the 'average' or 'mean,' offers insight into the central tendency of a distribution. For the Beta distribution, the expected value is determined by the parameters \(\alpha\) and \(\beta\).
Using integration, the expected value \(E(X)\) for a Beta random variable is calculated by:

• \( E(X) = \int_{0}^{1} x f(x) \, dx = \frac{\alpha}{\alpha + \beta} \)

This formula makes it evident that the mean is a ratio of the first parameter to the sum of both parameters, \(\alpha + \beta\).
The expected value is key as it summarizes the midpoint where most of the distribution's density is likely to arise, providing a valuable summary of the distribution's tendency.

Variance

Variance measures the spread or dispersion of a distribution, indicating how much the values can vary from the mean. In the context of the Beta distribution, variance depends heavily on both \(\alpha\) and \(\beta\).
To compute the variance \(\text{Var}(X)\), one must find the expected value of the square \(E(X^2)\) and use the formula:

• \( \text{Var}(X) = E(X^2) - [E(X)]^2 \)
• The result is: \( \text{Var}(X) = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)} \)

Variance is essential for understanding how widely spread the possible values are around the mean. In practical applications, knowing the variance helps in assessing the risk or reliability of different outcomes represented by the distribution.

Gamma Function

The Gamma Function \(\Gamma(n)\) is an extension of the factorial function to real numbers. It is highly important in the Beta distribution for calculating the Beta function.
The relationship between the Gamma function and factorial is \(\Gamma(n) = (n-1)!\) for positive integers \(n\).
For the Beta function \(\text{B}(\alpha, \beta)\), which is crucial in defining the Beta probability density function, it is expressed via Gamma functions as:

• \( \text{B}(\alpha, \beta) = \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} \)

The Gamma function helps tailor the PDF of a Beta distribution by allowing it to fit the required shape through \(\alpha\) and \(\beta\). This formulation ensures the total probability integrates to one, maintaining the fundamental property of probability distributions.