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When working with a uniform distribution, both the mean and variance offer insights into the expected average and the spread of data within a specific interval. For a continuous uniform distribution defined on the interval \(a, b\), the probability density function is constant, which suggests that every outcome in this range is equally likely.

To calculate the **mean** of a uniform distribution, you simply average the endpoints of the interval. The formula is \( \frac{a + b}{2} \). In this exercise, with \(a = 49.75\) and \(b = 50.25\), the mean becomes \( \frac{49.75 + 50.25}{2} = 50 \). This mean value represents the balance point, or center, of the distribution.

The **variance** measures how spread out the values are around the mean. For a uniform distribution, the variance is calculated using the formula \( \frac{(b-a)^2}{12} \). Here, the interval \(49.75, 50.25\) leads to a variance of \( \frac{(50.25 - 49.75)^2}{12} = 0.0208 \). Thus, you can expect a slight spread around the mean of 50 pounds.

The cumulative distribution function (CDF) of a uniform distribution provides insights into the probability that a random variable will be less than or equal to a certain value. It is particularly useful for understanding how probabilities accumulate over an interval.

For a uniform distribution over the interval \(a, b\), the CDF formula is \( F(x) = \frac{x - a}{b - a} \). This function transitions linearly from 0 at the start of the interval to 1 at the end. Specifically, within the range \(49.75 \leq x \leq 50.25\), the CDF can be expressed as \( F(x) = \frac{x - 49.75}{0.5} \).

This formulation allows one to determine the probability of observing a value below any specific point within the interval. By substituting different \(x\) values into this formula, you observe how the probability grows progressively larger as \(x\) approaches the upper bound.

Probability in the context of a uniform distribution is straightforward once the cumulative distribution function (CDF) is established. By using the CDF, you can find the likelihood of a random variable falling below a certain point.

For example, to compute \( P(X < 50.1) \), you utilize the CDF expression: \( F(x) = \frac{x - 49.75}{0.5} \). By inserting \(x = 50.1\) into this formula, the probability calculation becomes \( F(50.1) = \frac{50.1 - 49.75}{0.5} = 0.7 \).

This result, 0.7, signifies that there is a 70% chance that a randomly selected package weighs less than 50.1 pounds. This probability calculation exemplifies the practical application of the CDF in estimating the likelihood of outcomes within this distribution.