Problem 187 - Applied Statistics and Probability for Engineers

Question

(a) Calculate the mode, mean, and variance of the distribution for \(\alpha=3\) and \(\beta=1.4\) (b) Calculate the mode, mean, and variance of the distribution for \(\alpha=10\) and \(\beta=6.25\). (c) Comment on the difference in dispersion in the distribution from parts (a) and (b).A European standard value for a low-emission window glazing uses 0.59 as the proportion of solar energy that enters a room. Suppose that the distribution of the proportion of solar energy that enters a room is a beta random variable.

Step-by-Step Solution

Answer

For part (a), mean ≈ 0.6818, mode ≈ 0.8333, variance ≈ 0.0360; for part (b), mean ≈ 0.6154, mode ≈ 0.6316, variance ≈ 0.0134; part (b) has less dispersion.

Step 1: Understand Beta Distribution

The probability density function of a Beta distribution is defined by two parameters, \( \alpha \) and \( \beta \). The mean (expected value) of a beta distribution is given by \( \mu = \frac{\alpha}{\alpha + \beta} \), the mode is \( \frac{\alpha - 1}{\alpha + \beta - 2} \) for \( \alpha > 1 \) and \( \beta > 1 \), and the variance is \( \sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \). These formulas are essential for solving the problem when calculating these statistics for the given \( \alpha \) and \( \beta \) values.

Step 2: Calculate for \( \alpha = 3 \) and \( \beta = 1.4 \)

For \( \alpha = 3 \) and \( \beta = 1.4 \):* Mean: \( \mu = \frac{3}{3 + 1.4} = \frac{3}{4.4} \approx 0.6818 \)* Mode: \( \frac{3 - 1}{3 + 1.4 - 2} = \frac{2}{2.4} \approx 0.8333 \). Note that this is applicable given \( \alpha > 1 \) and \( \beta > 1 \).* Variance: \( \sigma^2 = \frac{3 \times 1.4}{(3 + 1.4)^2 \times (3 + 1.4 + 1)} = \frac{4.2}{4.4^2 \times 5.4} \approx 0.0360 \).

Step 3: Calculate for \( \alpha = 10 \) and \( \beta = 6.25 \)

For \( \alpha = 10 \) and \( \beta = 6.25 \):* Mean: \( \mu = \frac{10}{10 + 6.25} = \frac{10}{16.25} \approx 0.6154 \)* Mode: \( \frac{10 - 1}{10 + 6.25 - 2} = \frac{9}{14.25} \approx 0.6316 \).* Variance: \( \sigma^2 = \frac{10 \times 6.25}{(10 + 6.25)^2 \times (10 + 6.25 + 1)} = \frac{62.5}{16.25^2 \times 17.25} \approx 0.0134 \).

Step 4: Compare Distributions

The difference in dispersion can be observed from the variance calculations. The variance for \( \alpha=3 \) and \( \beta=1.4 \) is \( \approx 0.0360 \), indicating more dispersion, while for \( \alpha=10 \) and \( \beta=6.25 \) it is \( \approx 0.0134 \), indicating less dispersion. This suggests that the distribution with \( \alpha=10 \) and \( \beta=6.25 \) is more concentrated around the mean compared to the distribution with \( \alpha=3 \) and \( \beta=1.4 \).

Key Concepts

Probability Density Function

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 \([0,1]\).

Mean, Mode, and Variance

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.

Parameter Calculation

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.