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The probability calculation in a Weibull distribution helps us determine the likelihood that a random variable falls within certain numerical boundaries. If we consider a situation where we want to find the probability that the time before a person becomes infectious is between 1 and 2 days, we would use the cumulative distribution function (CDF) of the Weibull distribution.

For the given scenario, we utilize the parameters of the Weibull distribution: \( \alpha = 2.2 \), \( \beta = 1.1 \), and \( \gamma = 0.5 \). The cumulative distribution function is expressed as:\[ F(x) = 1 - e^{-\left(\frac{x-\alpha}{\beta}\right)^\gamma} \].

To calculate the probability for the interval \( P(1 < X < 2) \), we need to find the CDF values at \( x = 1 \) and \( x = 2 \), then subtract these numeric results:

• Compute \( F(1) \), which represents the probability up to 1 day.
• Compute \( F(2) \), which represents the probability up to 2 days.
• The probability \( P(1 < X < 2) \) is calculated as \( F(2) - F(1) \).

By performing these calculations, a specific figure numerically defines the likelihood of becoming infectious between the first and second days after exposure.

In the context of a Weibull distribution, the percentile calculation seeks to identify a particular value of \( X \) corresponding to a given cumulative probability. Specifically, figuring out the 90th percentile allows us to predict the time by which 90% of individuals will have become infectious.

This process involves solving the equation:\[ 1 - e^{-\left(\frac{x-\alpha}{\beta}\right)^\gamma} = 0.9 \]for \( x \). Here, the goal is to find the value of \( x \) for which the cumulative distribution function equals 0.9.

• This involves possibly using numerical methods or specialized statistical software to solve the equation.
• The result provides the threshold day by which 90% of the exposed individuals become infectious.

Understanding this percentile helps in planning and implementing containment protocols, as it highlights a significant time frame for most cases.

The mean and standard deviation offer insight into the central tendency and variability of the Weibull distribution, respectively. These statistics are crucial for understanding the spread and average time until infectiousness for those exposed to the disease.

• The mean \( (\mu) \) indicates the average time before infectiousness. It's calculated using:\[ \mu = \alpha + \beta \Gamma\left(1 + \frac{1}{\gamma}\right) \], where \( \Gamma \) represents the gamma function.
• The standard deviation \( (\sigma) \) gives us an idea of how much individual data points differ from the mean. It's calculated with:\[ \sigma = \beta \sqrt{\Gamma\left(1 + \frac{2}{\gamma}\right) - \left(\Gamma\left(1 + \frac{1}{\gamma}\right)\right)^2} \].

These calculations provide a fuller picture of the distribution, helping public health officials to estimate the variability and expected time from exposure to becoming infectious, thus aiding in health resource allocation and predicting outbreak patterns.