91Ó°ÊÓ

The mean life of an MRI machine, modeled by the Weibull distribution, provides an average indication of how long the machine is expected to operate before failure. The Weibull distribution is well-suited for life data analysis, often used in reliability and failure time analysis. The mathematical formula for the mean of a Weibull distribution is given by:\[\mu = \delta \Gamma(1 + 1/\beta)\]where:

• \(\mu\) is the mean or the expected value.
• \(\delta\) is the scale parameter, affecting the spread of the distribution.
• \(\beta\) is the shape parameter, determining the shape of the distribution.
• \(\Gamma\) is the gamma function, a sophisticated function that generalizes the factorial function to non-integer values.

In our problem, we have \(\delta = 500\) and \(\beta = 2\). By substituting these into the formula, we calculate \(\Gamma(1.5)\) which approximates to 0.8862. Thus, the mean is:\[\mu = 500 \times 0.8862 \approx 443.1 \text{ hours}\]This indicates that, on average, the MRI machine is expected to last about 443 hours.

The variance in the context of Weibull distribution describes the spread or dispersion of the life span of an MRI machine. It is calculated using the formula:\[\sigma^2 = \delta^2 \left( \Gamma(1 + 2/\beta) - \left( \Gamma(1 + 1/\beta) \right)^2 \right)\]where:

• \(\sigma^2\) is the variance.
• \(\delta\) and \(\beta\) are the scale and shape parameters, respectively.
• \(\Gamma\) represents the gamma function.

For our specific problem, the parameters \(\delta = 500\) and \(\beta = 2\) mean we need to calculate the two gamma functions. Previously, we found \(\Gamma(1.5) \approx 0.8862\). We also know \(\Gamma(2) = 1\).By substituting these values, we can compute the variance:\[\sigma^2 = 500^2 \times (1 - 0.8862^2) = 250000 \times (1 - 0.7859) \approx 26797.5 \text{ hours squared}\]This variance value indicates how much the lifespan of the MRIs can vary from the mean.

Understanding the probability of failure is critical in reliability engineering. It helps us estimate the likelihood that an MRI machine will fail before a particular time. For the Weibull distribution, this probability can be found using the cumulative distribution function (CDF):\[F(t) = 1 - e^{-(t/\delta)^\beta}\]where:

• \(F(t)\) is the cumulative probability of failure by time \(t\).
• \(e\) represents the exponential function.
• \(\delta\) and \(\beta\) are the scale and shape parameters of the distribution, respectively.

In the problem, we need to find the probability that the MRI fails before 250 hours. Using \(\delta = 500\) and \(\beta = 2\), the formula to calculate this becomes:\[F(250) = 1 - e^{-(250/500)^2} = 1 - e^{-0.25} \approx 1 - 0.7788 = 0.2212\]Therefore, the probability that the MRI will fail before 250 hours is about 22.12%. Knowing this helps in making informed maintenance and risk management decisions.