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When dealing with the Weibull distribution, probability calculation is essential to determine the likelihood of specific outcomes. In the given exercise, we are interested in finding two main probabilities: the probability that a magnetic disk lasts at least 500 hours, and the probability that it fails before 400 hours.

To calculate these probabilities, we use functions specific to the Weibull distribution. For at least 500 hours, we use the survival function, which indicates the probability of surviving beyond a certain point in time. Conversely, for failure before 400 hours, we employ the cumulative distribution function (CDF), which helps us understand the probability that a random variable is less than or equal to a certain value.

This calculation requires knowledge of the disk's life distribution, presented through parameters that are crucial for accurate results. Let's delve into the scale parameter next.

The scale parameter, often denoted as \( \eta \), is a critical component in the Weibull distribution. It essentially stretches or shrinks the distribution along the horizontal axis. In the context of reliability, like in our exercise involving magnetic disk life, the scale parameter influences the expected lifespan.

To find \( \eta \), we use the mean life formula of the Weibull distribution: \( \eta \cdot \Gamma(1+1/\beta) \). In this case, with a mean life of 600 hours and a shape parameter \( \beta = 0.5 \), solving for \( \eta \) gives us approximately 676.93.

This parameter allows us to apply the Weibull probability functions accurately, ensuring our calculations of survival and failure rates are correct. Next, let's understand the gamma function, which plays a role in finding \( \eta \).

The gamma function \( \Gamma(x) \) is a special mathematical function that extends the concept of factorial to real and complex numbers. It's important in various fields, including probability and statistics.

In the context of the Weibull distribution, the gamma function is used to calculate the mean when you don't have a simple integer shape parameter. When \( \beta = 0.5 \), the term \( 1 + 1/\beta = 1.5 \), leading us to the value \( \Gamma(1.5) \), which is approximately 0.8862.

This value is crucial for determining the scale parameter \( \eta \). In our exercise, it allowed us to solve the equation for \( \eta \) given the known mean life, making it possible to proceed with accurate probability calculations. Lastly, we explore the cumulative distribution function, vital for understanding probability in our scenario.

The cumulative distribution function (CDF) \( F(t) \) is an essential tool in probability and statistics. It represents the probability that a random variable is less than or equal to a specific value. In relation to Weibull distribution, the CDF formula is \( F(t) = 1 - e^{-(t/\eta)^\beta} \).

In the exercise, the CDF helps determine the probability of a disk failing before 400 hours. By inserting \( t = 400 \), \( \eta = 676.93 \), and \( \beta = 0.5 \) into the formula, we calculated a probability of approximately 0.539.

This means there's a 53.9% chance the disk will fail before reaching 400 hours, illustrating how valuable the CDF is in predicting real-world outcomes. Understanding the CDF not only contributes to precise probability calculations but also provides insights into the distribution of data over its range.