fter a quality training module was implemented, an evaluation was done to assess how well individuals accurately detected defective products. The evaluation involved having individuals examine a batch of products which included acceptable and defective products.

Given that a product is defective, the probability of correctly identifying the defect is denoted θθ. One individual examined a very large batch of products that included 1313 with defects. Of those 1313 defective products, they correctly identified 66 as defective and misidentified 77 as acceptable.

Assuming the number of defective correctly identified is Binomial(13,θBinomial(13,θ ) and a Beta(5.2,2.7)Beta(5.2,2.7) prior distribution, inputting values with 2 decimal points (e.g., 0.93):

What is the posterior expected value? 

What is the posterior standard deviation?  


We are given that the individual examined 13 defective products, correctly identified 6, and misidentified 7. The number of correctly identified defective products follows a binomial distribution:

\[
X \sim \text{Binomial}(n = 13, \theta)
\]

Additionally, we are provided with a **Beta prior** distribution:

\[
\theta \sim \text{Beta}(5.2, 2.7)
\]

### 1. **Posterior Distribution**

For Bayesian analysis, when using a Beta prior and a Binomial likelihood, the posterior distribution for \(\theta\) also follows a Beta distribution.

The prior is:

\[
\theta \sim \text{Beta}(\alpha_{\text{prior}} = 5.2, \beta_{\text{prior}} = 2.7)
\]

The likelihood, which is binomial, adds information to update our prior distribution. After observing the data, the updated posterior distribution becomes:

\[
\theta \mid X \sim \text{Beta}(\alpha_{\text{post}}, \beta_{\text{post}})
\]

Where the posterior parameters are updated as:

\[
\alpha_{\text{post}} = \alpha_{\text{prior}} + \text{number of successes}
\]
\[
\beta_{\text{post}} = \beta_{\text{prior}} + \text{number of failures}
\]

From the data, we have:

- Number of successes = 6 (correctly identified defective products)
- Number of failures = 7 (misidentified defective products)

Thus, the posterior parameters are:

\[
\alpha_{\text{post}} = 5.2 + 6 = 11.2
\]
\[
\beta_{\text{post}} = 2.7 + 7 = 9.7
\]

Therefore, the posterior distribution is:

\[
\theta \mid X \sim \text{Beta}(11.2, 9.7)
\]

### 2. **Posterior Expected Value**

The expected value of a Beta distribution \(\text{Beta}(\alpha, \beta)\) is given by:

\[
\mathbb{E}[\theta] = \frac{\alpha}{\alpha + \beta}
\]

Substituting the posterior values:

\[
\mathbb{E}[\theta] = \frac{11.2}{11.2 + 9.7} = \frac{11.2}{20.9} \approx 0.54
\]

### 3. **Posterior Standard Deviation**

The standard deviation of a Beta distribution \(\text{Beta}(\alpha, \beta)\) is given by:

\[
\text{SD}[\theta] = \sqrt{\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}}
\]

Substituting the posterior values:

\[
\text{SD}[\theta] = \sqrt{\frac{11.2 \times 9.7}{(11.2 + 9.7)^2 (11.2 + 9.7 + 1)}}
\]
\[
\text{SD}[\theta] = \sqrt{\frac{108.64}{20.9^2 \times 21.9}} = \sqrt{\frac{108.64}{436.81 \times 21.9}} = \sqrt{\frac{108.64}{9566.14}} \approx \sqrt{0.01136} \approx 0.1065
\]

Thus, the **posterior standard deviation** is approximately:

\[
\text{SD}[\theta] \approx 0.11
\]

### Final Answer:

- Posterior expected value of \(\theta\): \(\approx 0.54\)
- Posterior standard deviation of \(\theta\): \(\approx 0.11\)

Would you like further details or have any other questions?

### Related questions:
1. How does the prior distribution influence the posterior distribution?
2. What happens to the posterior if the prior distribution is more informative (e.g., smaller variance)?
3. How would the posterior distribution change if more defective products were tested?
4. How does the binomial likelihood combine with the beta prior mathematically?
5. How would one interpret the result of the posterior mean in a real-world context?

### Tip:
In Bayesian statistics, the more data you have, the less the prior influences the posterior distribution, making the posterior rely more on the observed data.

After a quality training module was implemented, an evaluation was done to assess how well individuals accurately detected defective products. The evaluation involved having individuals examine a batch of products which included acceptable and defective products. Given that a product is defective, the probability of correctly identifying the defect is denoted θθ. One individual examined a very large batch of products that included 13 with defects. Of those 13 defective products, they correctly identified 6 as defective and misidentified 7 as acceptable. Assuming the number of defective correctly identified is Binomial(13, θ) and a Beta(5.2, 2.7) prior distribution, inputting values with 2 decimal points (e.g., 0.93): What is the posterior expected value? What is the posterior standard deviation?

This problem explores Bayesian inference in the context of detecting defective products using a Beta-Binomial model. Learn how to calculate the posterior expected value and posterior standard deviation, applying concepts such as the binomial distribution and beta prior. Suitable for undergraduate statistics students.

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