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One of the key aspects when analyzing a Weibull distribution is determining the mean life of a device, like a computer processing unit (CPU). In this context, the mean life represents the average time until failure or breakdown of the CPU. The formula used to calculate the mean life from a Weibull distribution is:\[\mu = \delta \cdot \Gamma\left(1 + \frac{1}{\beta}\right)\]where:

• \(\delta\) is the scale parameter which influences the distribution's width, set at 900 hours in this problem.
• \(\beta\) is the shape parameter that dictates the distribution's form, given as 3.
• \(\Gamma\) represents the gamma function, a crucial mathematical function that generalizes factorial operations to real and complex numbers.

To solve this for the given CPU example:
1. We calculate \( \Gamma(1 + \frac{1}{3}) \), which is approximately 0.893.2. Substitute into the formula: \[ \mu = 900 \times 0.893 \approx 803.7 \text{ hours} \]Thus, the mean life of the CPU is approximately 803.7 hours. This means on average, the device will fail around this time.

In statistics, variance is a measure of the dispersion or spread of a set of values. For a Weibull distribution, calculating variance helps understand how much the CPU life deviates from the mean life. The formula to compute variance \(\sigma^2\) for a Weibull distribution is:\[\sigma^2 = \delta^2 \left( \Gamma\left(1 + \frac{2}{\beta}\right) - \Gamma^2\left(1 + \frac{1}{\beta}\right) \right)\]In this exercise, we perform the following steps to determine the variance:
1. Calculate \( \Gamma\left(1 + \frac{2}{\beta}\right) \). For \(\beta = 3\), this gives us \(\Gamma(1 + \frac{2}{3}) \approx 1.329\).2. Note that \( \Gamma\left(1 + \frac{1}{\beta}\right) = 0.893 \) from the previous mean life calculation.3. Substitute the values into the variance formula: \[ \sigma^2 = 900^2 \left(1.329 - 0.893^2\right) = 900^2 \times 0.532 = 429,660 \text{ hours}^2 \]This variance tells us how widely the life spans of the CPUs might vary when compared to the mean. Larger variance indicates a wider range of lifetimes around the average.

The probability of failure quantifies the likelihood that a CPU will fail before a specific time. To calculate this using a Weibull distribution, we employ the cumulative distribution function (CDF), which is:\[F(x) = 1 - e^{-(x/\delta)^\beta}\]Here, we want to find the probability that the CPU will fail before 500 hours:

• \(x\): Desired time for failure probability (500 hours).
• \(\beta\): Shape parameter (3).
• \(\delta\): Scale parameter (900 hours).

Follow these steps:
1. Compute \((500/900)^3\). This is approximately 0.1715.2. Substitute into the CDF formula: \[ F(500) = 1 - e^{-0.1715} \approx 1 - 0.8424 = 0.1576 \]This result, 0.1576, is the probability that the CPU will fail before reaching 500 hours. Or, you could say there is a about 15.76% chance of failure before this time.