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A continuous probability distribution describes the likelihood that a random continuous variable falls within a certain range of values. In a continuous distribution, the variable can take any value within a specified range. The distribution is typically depicted by a curve or a graph where the area under the curve between two points represents the probability that the random variable falls within that interval.

Uniform distribution is an example of a continuous probability distribution. In this type of distribution, every outcome within the range is equally likely. Imagine if you roll a perfectly fair die with decimal points allowed between numbers; each face would show up with the same likelihood, forming a perfect rectangle if plotted on a graph. In our exercise, the flight time from 60 to 70 minutes is uniformly distributed, meaning that the probability of the flight taking exactly 62, 65, or 69 minutes is exactly the same.

• The graph of a uniform distribution is a rectangle.
• The height of the rectangle is the same across the range.
• The height is determined by the reciprocal of the range.

The mean of a uniform distribution is the central point, the balance or average value of the distribution. It is calculated by taking the average of the minimum and maximum values of the range. This mean tells us the expected flight time if flights are randomly sampled.

For a uniform distribution that ranges from value \(a\) to \(b\), the formula for the mean is\[\mu = \frac{a + b}{2}\]In our problem, the flight time is uniformly distributed between 60 and 70 minutes. Therefore, calculating the mean involves adding the smallest and largest values (60 and 70) and dividing by 2:

• \( \mu = \frac{60 + 70}{2} = 65 \)

This mean indicates that the central expected flight time is 65 minutes.

The variance of a uniform distribution quantifies the spread or how far the values of the distribution are from the mean. This is important as it tells us about the variability or consistency of the flight times.

For a continuous uniform distribution with minimum value \(a\) and maximum value \(b\), the formula for the variance is\[\sigma^2 = \frac{(b - a)^2}{12}\]Using the current exercise's values where \(a = 60\) and \(b = 70\), we can find the variance as follows:

• \(\sigma^2 = \frac{(70 - 60)^2}{12} = \frac{100}{12} \approx 8.33\)

This result indicates that while the mean flight time is steady at 65 minutes, individual flight times may vary moderately around this average.

The probability calculation in uniform distribution involves finding the likelihood that the flight time falls within a particular interval. Since the distribution is uniform, probability can be determined by calculating the area under the probability density function (PDF).

For this distribution, the PDF is constant across the interval, and the height is \(0.1\) because the probability over the entire interval of 60 to 70 minutes must sum to 1.

• Probability flight time is less than 68 minutes: Since the interval length from 60 to 68 is 8 minutes, the probability can be calculated as: \[ P(X < 68) = 8 \times 0.1 = 0.8 \]
• Probability flight time is more than 64 minutes: The interval from 64 to 70 is 6 minutes. Hence: \[ P(X > 64) = 6 \times 0.1 = 0.6 \]

Understanding these probabilities tells us how likely certain flight times are, allowing us to better anticipate and plan for these scenarios.