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A Cumulative Distribution Function (CDF) is an essential tool in statistics, especially when dealing with continuous probability distributions like the gamma distribution. It gives us the probability that a random variable takes on a value less than or equal to a specific point.
For a gamma distribution characterized by parameters shape \( \theta \) and scale \( \beta \), the CDF can be used to find probabilities like \( P(X \leq x) \), which is the probability that the lifetime of transistors is less than or equal to \( x \) weeks. In our example, to find the probability that a transistor lasts less than 24 weeks, we calculate \( P(X < 24) \) using the gamma CDF.

Once we have the CDF values for specific time points, like 12 weeks or 24 weeks, we can easily compute the probability of the transistor lasting between a range, such as 12 to 24 weeks, by subtracting the CDF value at 12 weeks from that at 24 weeks. Statistical software or gamma distribution tables typically provide these CDF values.

In a gamma distribution, the behavior and characteristics of the distribution are largely determined by its shape \( \theta \) and scale \( \beta \) parameters.
To find these parameters, we rely on the relationship between the mean \( \mu \) and standard deviation \( \sigma \) of the distribution. Given that \( \mu = \theta \beta \) and \( \sigma = \sqrt{\theta} \beta \), we can solve these to find the values of \( \theta \) and \( \beta \) when the mean is 24 weeks and the standard deviation is 12 weeks.

In this specific exercise, by solving \( \theta \beta = 24 \) and \( \sqrt{\theta} \beta = 12 \) simultaneously, we determine the shape parameter \( \theta = 4 \) and the scale parameter \( \beta = 6 \). These parameters are critical as they allow us to describe the distribution fully and perform additional calculations, like finding the cumulative distribution function values or percentiles.

Percentiles are particularly useful in understanding how a value compares within a distribution. The 99th percentile, for example, represents a value below which 99% of the data fall.
To calculate a percentile in a gamma distribution, we use the inverse of the cumulative distribution function (inverse CDF). Suppose we want the 99th percentile of the lifetime distribution of transistors. In that case, we find a value \( X_{99} \) such that \( P(X < X_{99}) = 0.99 \). This involves using statistical software or a calculator that supports the gamma distribution's inverse CDF.

Percentile calculations can help set performance benchmarks. For instance, if a transistor must perform better than 99% of all others, the 99th percentile value tells us the minimum performance criteria it needs to meet.

The median of a distribution represents the middle value, where half the observations are below this point, and half are above. Unlike symmetric distributions such as the normal distribution, where the median equals the mean, the gamma distribution's median can differ from the mean due to its skewness.
To find the median for the gamma distribution, we look for a value \( X \) such that \( P(X < \text{median}) = 0.5 \). This can be done using the gamma distribution's inverse CDF. In this exercise, following such a calculation allows us to determine if the median lifetime for the transistors is less than the mean, confirming whether the distribution is skewed.

Understanding the median is fundamentally important as it provides another way to assess the central tendency of data, especially in skewed distributions where the mean might not be a representative measure.