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Uniform distribution is one of the simplest types of probability distributions. It describes a situation where all outcomes in a given interval are equally likely. In our case, the net weight of the packaged chemical herbicide is uniformly distributed between 49.75 and 50.25 pounds.

Key features of uniform distribution include:

• All intervals of the same length within the distribution's range are equally probable.
• It can be either continuous or discrete, though our example is continuous.
• The graph of a continuous uniform distribution resembles a rectangle.

This means the height of this rectangle represents the constant probability density, making it straightforward to calculate probabilities and other statistics.

The cumulative distribution function, or CDF, provides the probability that a random variable will take a value less than or equal to a particular number. It essentially builds up probabilities as you move through a distribution. For the uniform distribution, the CDF is a linear function because probabilities accumulate at a constant rate.

In mathematical terms for a continuous uniform distribution between two values, \(a\) and \(b\), the CDF, \(F(x)\) is defined as:

• \(F(x) = 0\) if \(x < a\)
• \(F(x) = \frac{x-a}{b-a}\) if \(a \leq x < b\)
• \(F(x) = 1\) if \(x \geq b\)

For our specific interval of 49.75 to 50.25 pounds, the CDF creates a clear picture of how the probability is spread across the possible weight values.

In statistics, the mean is the average, providing a central value for a set of numbers. For a continuous uniform distribution, the mean is straightforwardly the midpoint of the interval. So, for our herbicide weight, which is uniformly distributed from 49.75 to 50.25 pounds, the mean weight \( \mu \) is: \[ \mu = \frac{49.75 + 50.25}{2} = 50.00 \]

Variance, on the other hand, measures the dispersion around the mean. In simple terms, it tells us how much the values spread out. For a uniform distribution \( U(a, b) \), the variance \( \sigma^2 \) is calculated as: \[ \sigma^2 = \frac{(b - a)^2}{12} \] For our example: \[ \sigma^2 = \frac{(50.25 - 49.75)^2}{12} = 0.0208 \]
This result indicates a low variance, meaning that the weights are tightly clustered around the mean of 50 pounds.

Probability calculation allows us to figure out the likelihood of certain outcomes within given constraints. With a uniform distribution, this becomes relatively simple due to the equal likelihood of results. We use the cumulative distribution function (CDF) to find probabilities for a specified range.

In our exercise, we calculated the probability that the net weight is less than 50.1 pounds by evaluating the CDF at that point. Since 50.1 is within our interval of 49.75 to 50.25, use: \[ F(50.1) = \frac{50.1 - 49.75}{0.5} = 0.7 \]
This calculation tells us that there's a 70% chance a given package weighs less than 50.1 pounds, providing practical insight into the distribution's behavior in everyday terms.