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The Beta Distribution is a versatile distribution defined on the interval [0, 1], making it suitable for modeling random variables that represent proportions or probabilities. This distribution is characterized by two shape parameters, denoted as \(\alpha\) and \(\beta\). These parameters control the "shape" of the distribution, determining how skewed it is and where the mode (peak) occurs.
- If \(\alpha = \beta = 1\), the distribution is uniform across the interval.- Increasing \(\alpha\) while keeping \(\beta < \alpha\) skews the distribution towards 1, resulting in a higher chance of observing values close to 1.- Similarly, if \(\beta > \alpha\), the distribution is skewed towards 0.
In the context of the given exercise, the random variable \(X\), representing the proportion of time patients wait at the emergency department, follows a Beta distribution with parameters \(\alpha = 10\) and \(\beta = 1\). This indicates a distribution heavily skewed towards 1, meaning that patients are likely to spend most of their time waiting.

The Cumulative Distribution Function (CDF) of a random variable describes the probability that the variable will take a value less than or equal to a specific value. For a Beta distribution, the CDF can be denoted as \(F(x)\), reflecting the cumulative probability up to \(x\) within the interval [0, 1].
To find probabilities such as \(P(X > 0.9)\) or \(P(X < 0.5)\), we utilize the CDF as follows:

• \(P(X > 0.9) = 1 - F(0.9)\): This gives the probability that \(X\) is greater than 0.9.
• \(P(X < 0.5) = F(0.5)\): This provides the probability that \(X\) is less than 0.5.

The CDF for the \(\text{Beta}(10, 1)\) distribution is calculated using statistical software or reference tables. Given the distribution's skewness, probabilities such as \(P(X > 0.9)\) and \(P(X < 0.5)\) will be small and large, respectively, due to a high concentration of values near 1.

Calculating the mean and variance of a Beta distribution is straightforward and provides valuable insights into the data's central tendency and spread.
- **Mean**: The mean or expected value of a Beta distribution \(\text{Beta}(\alpha, \beta)\) is given by the formula \(\frac{\alpha}{\alpha + \beta}\). It represents the center of mass of the distribution. For \(\alpha = 10\) and \(\beta = 1\), the mean is \(\frac{10}{11}\), indicating a high average proportion, due to the skewness towards larger values.- **Variance**: The variance of the distribution, which measures the spread around the mean, is calculated by \(\frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\). For our parameters, it becomes \(\frac{10}{11^2 \times 12}\), quantifying the variability in the proportion of time spent waiting. This low variance highlights the consistency of the time proportion across cases.