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In Weibull distribution, parameter estimation is crucial for understanding the behavior of a product's lifespan. The two main parameters are the shape parameter, \( \beta \), and the scale parameter, \( \delta \).

• **Shape parameter (\( \beta \))**: Determines how the failure rate changes over time. A \( \beta \) less than 1 indicates a decreasing failure rate, \( \beta \) equal to 1 indicates a constant failure rate, and \( \beta \) greater than 1 indicates an increasing failure rate.
• **Scale parameter (\( \delta \))**: Influences the scale of the data, often related to the average life of the items.

For this exercise, we have \( \beta = 3 \) and \( \delta = 900 \) hours. This configuration typically indicates an increasing failure rate, suggesting that the risk of failure grows as the CPU approaches its lifespan. Accurate estimation of these parameters allows for better predictions and planning regarding replacements and maintenance schedules. Understanding these parameters provides the groundwork for further calculations such as mean life and failure probability.

Calculating the mean life of a CPU using the Weibull distribution helps us determine the expected duration the CPU will function before failing. The formula to find the mean life \( E(X) \) is:\[E(X) = \delta \cdot \Gamma(1 + 1/\beta) \]Where \( \Gamma \) is the gamma function. In our case:

• \( \delta = 900 \)
• \( \beta = 3 \)
• \( \Gamma(1 + 1/3) \approx 0.89297951 \)

Substituting these values, we compute the mean:\[E(X) = 900 \times 0.89297951 \approx 803.68 \text{ hours}\]This tells us that, on average, a CPU will last around 803.68 hours before it is likely to fail. This information is pivotal for product life cycle management and ensuring that replacements and spare parts are appropriately managed.

Variance provides insight into the variability of CPU lifespans. It shows how much the lifetimes differ from the mean. For a Weibull distribution, the variance \( \text{Var}(X) \) is given by:\[\text{Var}(X) = \delta^2 \left( \Gamma(1 + 2/\beta) - (\Gamma(1 + 1/\beta))^2 \right)\]In our problem:

• \( \delta = 900 \)
• \( \Gamma(1 + 2/3) \approx 1.160529 \)
• \( \Gamma(1 + 1/3) \approx 0.89297951 \)

We substitute these values into the variance formula:\[\text{Var}(X) = 900^2 \times (1.160529 - 0.797415) \approx 186,676 \text{ hours squared}\]A higher variance indicates greater variability in lifespans. This value allows manufacturers and managers to assess the consistency of the CPU's life and make decisions regarding quality and reliability enhancements.

Understanding the failure probability of a CPU before a certain time period is vital for reliability calculations. The cumulative distribution function (CDF) for the Weibull distribution helps in calculating this probability. The equation is:\[F(t) = 1 - e^{-(t/\delta)^\beta}\]For our CPU, to find the probability of failure before 500 hours:

• \( t = 500 \)
• \( \delta = 900 \)
• \( \beta = 3 \)

First, compute \((500/900)^3 \approx 0.17146776\). Then, calculate:\[F(500) = 1 - e^{-0.17146776} \approx 1 - 0.842 \approx 0.158\]This result means there's a 15.8% chance of the CPU failing before it reaches 500 hours of operation. This probability helps in assessing risks related to product durability and planning for maintenance or warranty terms.