10 Suppose that a system contains a certain type of component whose time, in years, to fallure is given by F. The random variable 7 is modeled nicely by the exponential distribution with mean time to failure beta = 5 If 5 of these components are installed in different systems, what is the probability that at least 2 are still functioning at the end of 8 years?

(0) It is known, from previous data, that the length of time in manths between customer complaints about a certain product is a gamma distribution with a = 2 and beta = 4 Changes were made to tighten quality control requirements. Following these , 20 months passed before the first complaint. Does it appear as if the quality control tightening was effective? (Given: F(5; 2) = 0.96 )

Let’s break down each part of this problem.

### Problem (a): Exponential Distribution

We have:
- A system with components whose time to failure \( F \) follows an exponential distribution with mean \( \beta = 5 \) years.
- There are 5 independent components installed, and we are asked to find the probability that at least 2 of them are still functioning after 8 years.

#### Step 1: Probability of a Single Component Functioning for 8 Years

The exponential distribution has a probability density function \( f(t) = \frac{1}{\beta} e^{-t/\beta} \) and a cumulative distribution function \( F(t) = 1 - e^{-t/\beta} \).

1. The probability that a component is still functioning after 8 years (i.e., \( P(T > 8) \)) is:
   \[
   P(T > 8) = 1 - F(8) = e^{-8/\beta} = e^{-8/5}
   \]
   
2. Calculating \( e^{-8/5} \):
   \[
   P(T > 8) \approx e^{-1.6} \approx 0.2019
   \]

#### Step 2: Probability That at Least 2 Components Are Functioning

Let \( X \) represent the number of components functioning after 8 years. \( X \) follows a binomial distribution, \( X \sim \text{Binomial}(n = 5, p = 0.2019) \), where \( p = 0.2019 \) is the probability of a single component still working after 8 years.

We need \( P(X \geq 2) \), which is:
\[
P(X \geq 2) = 1 - P(X = 0) - P(X = 1)
\]

Using the binomial probability formula \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \):

1. **Calculate \( P(X = 0) \):**
   \[
   P(X = 0) = \binom{5}{0} (0.2019)^0 (1 - 0.2019)^5 = (0.7981)^5 \approx 0.3277
   \]

2. **Calculate \( P(X = 1) \):**
   \[
   P(X = 1) = \binom{5}{1} (0.2019)^1 (0.7981)^4 \approx 5 \times 0.2019 \times 0.4066 \approx 0.4105
   \]

3. **Calculate \( P(X \geq 2) \):**
   \[
   P(X \geq 2) = 1 - P(X = 0) - P(X = 1) \approx 1 - 0.3277 - 0.4105 \approx 0.2618
   \]

Thus, the probability that at least 2 components are still functioning after 8 years is approximately **0.2618**.

---

### Problem (b): Gamma Distribution for Complaint Time

For this part:
- The time between customer complaints follows a gamma distribution with parameters \( \alpha = 2 \) and \( \beta = 4 \).
- Following quality control changes, 20 months passed before a complaint.
- We need to determine if this 20-month interval suggests an effective change.

1. **Expected Time Between Complaints (Mean)**: For a gamma distribution, the mean \( \mu = \alpha \cdot \beta = 2 \cdot 4 = 8 \) months.

2. **Probability of Observing a Gap of 20 Months or More**: We calculate \( P(X \geq 20) \) using the cumulative distribution function (CDF) for the gamma distribution.

Since \( P(X \leq 5) = 0.96 \) is given, we can infer that 5 months is a very short interval relative to typical gaps, while 20 months would be unusually long, suggesting that quality control was likely effective.

---

Would you like to go deeper into each calculation or have further clarifications? 

### Further Questions to Explore:
1. How would the result in part (a) change if we increased the mean time to failure?
2. What happens to the probability in part (a) if the components are not independent?
3. How does a gamma distribution differ from an exponential distribution, and when is each used?
4. Can we quantify how "effective" quality control was with more data?
5. How do the parameters \( \alpha \) and \( \beta \) influence the shape of the gamma distribution?

### Tip:
Gamma and exponential distributions are related: the exponential distribution is a special case of the gamma distribution when \( \alpha = 1 \).

Suppose that a system contains a certain type of component whose time, in years, to failure is given by F. The random variable is modeled nicely by the exponential distribution with mean time to failure beta = 5. If 5 of these components are installed in different systems, what is the probability that at least 2 are still functioning at the end of 8 years? It is known, from previous data, that the length of time in months between customer complaints about a certain product is a gamma distribution with a = 2 and beta = 4. Changes were made to tighten quality control requirements. Following these, 20 months passed before the first complaint. Does it appear as if the quality control tightening was effective? (Given: F(5; 2) = 0.96)

Explore an advanced probability problem involving exponential and gamma distributions. Calculate the probability that at least two of five components will function beyond 8 years and assess quality control effectiveness over a 20-month period without complaints. Ideal for college students studying probability and distributions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation