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Chapter 4: Problem 132

Proper blending of fine and coarse powders prior to copper sintering is essential for uniformity in the finished product. One way to check the homogeneity of the blend is to select many small samples of the blended powders and measure the proportion of the total weight contributed by the fine powders in each. These measurements should be relatively stable if a homogeneous blend has been obtained. a. Suppose that the proportion of total weight contributed by the fine powders has a beta distribution with \(\alpha=\beta=3 .\) Find the mean and variance of the proportion of weight contributed by the fine powders. b. Repeat part (a) if \(\alpha=\beta=2\) c. Repeat part (a) if \(\alpha=\beta=1\) d. Which of the cases-parts \((\underline{a}),(\underline{b}),\) or \((\underline{c})-\) yields the most homogeneous blending?

Short Answer

Expert verified
Case a (\(\alpha=\beta=3\)) yields the most homogeneous blending with the lowest variance (\(\frac{1}{28}\)).

Step by step solution

01

Define Mean and Variance for Beta Distribution

For a beta distribution with parameters \( \alpha \) and \( \beta \), the mean \( \mu \) is given by \( \mu = \frac{\alpha}{\alpha + \beta} \) and the variance \( \sigma^2 \) is given by \( \sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \).
02

Calculate for Case a (\(\alpha=\beta=3\))

Using \( \alpha=3 \) and \( \beta=3 \), compute\[ \mu = \frac{3}{3+3} = \frac{1}{2} \]and the variance:\[ \sigma^2 = \frac{3 \times 3}{(3 + 3)^2 (3 + 3 + 1)} = \frac{9}{36 \times 7} = \frac{1}{28}. \]
03

Calculate for Case b (\(\alpha=\beta=2\))

Using \( \alpha=2 \) and \( \beta=2 \), compute\[ \mu = \frac{2}{2+2} = \frac{1}{2} \]and the variance:\[ \sigma^2 = \frac{2 \times 2}{(2 + 2)^2 (2 + 2 + 1)} = \frac{4}{16 \times 5} = \frac{1}{20}. \]
04

Calculate for Case c (\(\alpha=\beta=1\))

Using \( \alpha=1 \) and \( \beta=1 \), compute\[ \mu = \frac{1}{1+1} = \frac{1}{2} \]and the variance:\[ \sigma^2 = \frac{1 \times 1}{(1 + 1)^2 (1 + 1 + 1)} = \frac{1}{4 \times 3} = \frac{1}{12}. \]
05

Determine Most Homogeneous Blend

Homogeneity is associated with lower variance in the proportions. Compare the variances: \( \frac{1}{28} \) from case a, \( \frac{1}{20} \) from case b, and \( \frac{1}{12} \) from case c. The smallest variance \( \frac{1}{28} \) is from case a, indicating the most uniform blending.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean and Variance
The mean and variance are fundamental concepts in statistics used to describe the characteristics of a probability distribution. When dealing with a Beta distribution, which is commonly used to model proportions, understanding how to calculate these values is crucial.

For a Beta distribution with parameters \(\alpha\) and \(\beta\), the mean (denoted as \(\mu\)) is calculated using the formula:
  • \(\mu = \frac{\alpha}{\alpha + \beta}\).
The mean represents the expected proportion of fine powders in the blend, a critical measure of central tendency. Regardless of the parameters used, in our exercise, this reported value was always \(\frac{1}{2}\), implying balance in the mixture.

The variance (denoted as \(\sigma^2\)) provides insight into the spread or variability of the distribution. It is calculated using:
  • \(\sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\).
This value helps determine the consistency of the mixture, with smaller values indicating more stability in the blend. In our cases, variance decreases as the \(\alpha\) and \(\beta\) parameters increase, reflecting a more homogeneous blend.
Homogeneous Blending
Homogeneous blending refers to the uniform distribution of components within a mixture. In the context of our exercise, the objective is to ensure that fine powders are evenly distributed when mixing with coarse powders.

A homogeneous blend will produce consistent measurements of the proportion of fine powders across different samples. This consistency is quantitatively reflected by lower variance in the distribution. When variance is minimized, it indicates minimal fluctuation in the sample proportions, which is desirable.

In our exercise, with parameters \(\alpha = \beta = 3\), the calculated variance was \(\frac{1}{28}\), the smallest among our compared scenarios. This signals that when both \(\alpha\) and \(\beta\) are larger but equal, the resulting blend is the most uniform, exhibiting fewer variations in sample measurements.
Parameter Estimation
Parameter estimation in statistical distributions is the process of determining the values of parameters such as \(\alpha\) and \(\beta\) that characterize the distribution. Accurate parameter estimation allows us to model real-world processes effectively.

In the context of our exercise, we needed to estimate these parameters to determine the mean and variance, key to assessing blend uniformity. Each pair of parameters \(\alpha\) and \(\beta\) affects these two properties.

To emulate a homogeneous mixture, the selection of parameters \(\alpha\) and \(\beta\) must be precise. The cases with different sets of equal \(\alpha\) and \(\beta\) values demonstrated how these parameters directly influence homogeneity. By computing the mean and variance for each parameter pair, we could compare their effects on blend consistency, as seen when different parameter values produced different variances. The goal is to find parameters that yield the smallest variance, signifying optimal homogeneity.

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Most popular questions from this chapter

The median of the distribution of a continuous random variable \(Y\) is the value \(\phi_{5}\) such that \(P\left(Y \leq \phi_{5}\right)=0.5 .\) What is the median of the uniform distribution on the interval \(\left(\theta_{1}, \theta_{2}\right) ?\)

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