91Ó°ÊÓ

The mean, also known as the average, offers a central value for a distribution. In a continuous uniform distribution, where every value between two points is equally likely, calculating the mean is straightforward. The formula to find the mean of a uniform distribution defined on an interval \(a\) to \(b\) is \((a + b) / 2\).
For instance, considering an interval from \(49.75\) to \(50.25\), substitute the values into the formula:
\[ \text{Mean} = \frac{49.75 + 50.25}{2} = 50. \] This result shows that the mean weight of the packages is 50 pounds. Calculating the mean provides insights into the expected or average net weight inside the packages.

Variance is a measure that expresses how spread out the values in a distribution are. In the context of a uniform distribution, it shows how much each potential outcome differs from the mean. The formula for the variance in a continuous uniform distribution defined over \(a, b\) is \((b-a)^2/12\).
Applying this to our interval \(49.75, 50.25\):- Calculate the difference: \(b - a = 50.25 - 49.75 = 0.5\)- Then apply the variance formula:
\[ \text{Variance} = \frac{(0.5)^2}{12} = \frac{0.25}{12} \approx 0.0208. \]
The variance of approximately \(0.0208\) reflects a relatively small spread, indicating that the weights of the packages are closely packed around the mean.

The Cumulative Distribution Function (CDF) is a critical concept that tells us the probability that a random variable takes a value less than or equal to a certain point. For a continuous uniform distribution over an interval \(a\) to \(b\), the CDF is defined piecewise:
- \(F(x) = 0\) for \(x < a\) - \(F(x) = \frac{x - a}{b-a}\) for \(a \leq x \leq b\)- \(F(x) = 1\) for \(x > b\)
In this particular problem, when \(x\) is between \(49.75\) and \(50.25\), the CDF \(F(x)\) is computed as follows:
\[ F(x) = \frac{x - 49.75}{0.5},\] which means the CDF linearly increases from 0 to 1 as \(x\) travels from \(49.75\) to \(50.25\). This function helps us understand how probabilities accumulate over the distribution's range.

Calculating probabilities using the Cumulative Distribution Function (CDF) is a valuable application of statistics. It reveals how likely a random variable is to fall below a certain point. To find \(P(X < 50.1)\), we plug \(50.1\) into the CDF:
\[ F(50.1) = \frac{50.1 - 49.75}{0.5} = 0.7. \]
This tells us there is a 70% chance that the net weight of a random package is less than 50.1 pounds. Probabilities derived from the CDF are exceedingly useful in predicting outcomes and making decisions based on statistical data.