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When dealing with uniform distributions, the probability density function (PDF) is pretty simple and straightforward. Think of it like a fair game: every possible outcome between two points is equally probable. For this particular case of flight times between 2 hours (120 minutes) and 2 hours, 20 minutes (140 minutes), each minute in that range is just as likely as the next.
To visualize this, the PDF is a horizontal line on a graph. This line stretches from our starting point, 120 minutes, to the ending point, 140 minutes. Because we have 20 equally likely minutes (from 120 to 140), the height of this line is calculated as \[ \frac{1}{b-a} \] or \[ \frac{1}{140-120} = \frac{1}{20}. \] This means that each minute has a probability of \( \frac{1}{20} \). Outside this range, the value of the PDF drops to zero, making sure we only consider times within the bounds we've set.

In statistics, the expected value is like the average or the mean of a probability distribution. For a uniform distribution, it’s easy to calculate. You're basically looking at the middle point between two values.
Here, our flight times span from 120 to 140 minutes. To find that middle point, we calculate \[ E(X) = \frac{a+b}{2} \]. So by plugging in our numbers, the expected flight time is \[ \frac{120+140}{2} = 130 \] minutes. This expected value, 130 minutes or 2 hours and 10 minutes, gives you an idea of the average flight time you might expect if the flights followed this pattern consistently. It helps in making predictions and planning based on the most likely average outcome.

Flight time variability can be modeled using different types of probability distributions. In our exercise, the uniform distribution was chosen.
The uniform distribution is especially handy when you know the minimum and maximum hours but aren't sure about what happens in between. Here, any minute from 120 to 140 is equally likely.
Let's dive into a couple of scenarios. Firstly, a flight is considered on schedule if it lands not more than 5 minutes past the scheduled time, i.e., by 125 minutes. The chance it meets this criterion is \( P(X \leq 125) = \frac{125-120}{140-120} = \frac{5}{20} = 0.25 \).Then, if you're wondering whether it might arrive more than 10 minutes late, you look at the chances it takes over 130 minutes. This is \( P(X > 130) = 1 - P(X \leq 130) = 1 - \frac{10}{20} = 0.5 \). That’s a 50% chance of being over 10 minutes late. These calculations show both punctuality and delays' probabilities, helping us better understand what to expect from the airline's schedule.