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The Cumulative Distribution Function (CDF) is a fundamental tool for understanding probability distributions. In the context of a beta distribution, the CDF helps us understand the probability that a random variable, say X, will take a value less than or equal to a specific number within the interval [0, 1].
For a beta distribution with parameters \(\alpha\) and \(\beta\), the CDF of \(X\) up to a point \(x\) is calculated using the beta function. However, in practice, we often use tools or tables, as seen in the exercise, to determine these values.
To find \(P(X < 0.25)\) with \(X \sim \text{Beta}(1, 4.2)\), the CDF is approximately 0.6762. This means there is about a 67.62% likelihood that \(X\) falls below 0.25.

Calculating the mean and variance is crucial for describing a distribution comprehensively. For the beta distribution, these terms directly relate to its parameters \(\alpha\) and \(\beta\).
**Mean**: The mean of a beta distribution is determined by the formula \(\frac{\alpha}{\alpha + \beta}\). This formula highlights how the shape parameters influence the center of the distribution. In our particular case with \(\alpha = 1\) and \(\beta = 4.2\), the mean is calculated as \(\frac{1}{1+4.2} = 0.1923\). This indicates that the expected value of \(X\) is around 0.1923.
**Variance**: The variance formula provides insight into the spread of the distribution: \(\frac{\alpha \cdot \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}\). For the given parameters \(\alpha = 1\) and \(\beta = 4.2\), the variance computes to roughly 0.0322, indicating a relatively concentrated spread around the mean.

Calculating probabilities for certain intervals can uncover the likelihood of events falling within specified ranges. For a beta distribution, this often involves computing or using complementary probabilities through the CDF.
To determine \(P(0.5 < X)\), we'll first find its complementary probability, which involves \(P(X \leq 0.5)\). Using the CDF, this probability is found to be approximately 0.9375. Hence, \(P(0.5 < X) = 1 - P(X \leq 0.5) = 1 - 0.9375 = 0.0625\).
What this means is there's about a 6.25% chance that \(X\) is greater than 0.5, indicating the rarity of observing values in this upper portion of the distribution.