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When discussing uniform distribution, the Probability Density Function (PDF) serves as an essential concept. In the context of a continuous uniform distribution, the PDF is remarkably simple. It is defined mathematically as:

• If the random variable \(X\) lies between \(A\) and \(B\), then the PDF is expressed as \(f(x) = \frac{1}{B-A}\).
• For values of \(x\) outside this range, \(f(x) = 0\).

This function essentially tells you how likely it is for the variable to take on a particular range of values. For example, if \(A = 25\) and \(B = 35\), substituting these into our formula gives \(f(x) = \frac{1}{35-25} = \frac{1}{10}\) for each unit interval between 25 and 35.
This means the distribution is consistent and does not favor any interval over another. So effectively, if you think about it, this forms a flat, rectangular shape when graphed, which is why it is also referred to as a rectangular distribution.

In the realm of continuous distributions, such as our uniform distribution, probability is concerned with ranges rather than specific values. With a uniform distribution, "where" within that range doesn’t matter because each point on the interval is equally likely. So, how do we express probability here?
Since the total area beneath the PDF curve equals 1, representing 100% probability, the probability of \(X\) falling within any part of the interval is proportional to the length within that range.

• For instance, if we want to know the probability that the prep time exceeds 33 minutes, we would calculate the area from 33 to 35.
• Mathematically, this would be \((35 - 33) \times \frac{1}{10} = 0.2\).

Similarly, if you are measuring a specific subset range, like falling within 2 minutes of the mean preparation time, it simply involves multiplying the length of that range by the PDF. By understanding this, you can compute probabilities for any segment within the interval \(A\) to \(B\).

The Mean, denoted as \(\mu\), provides a central value for our distribution. For a uniform distribution, calculating the mean is straightforward. It is the midpoint of \(A\) and \(B\). This can be handy in quickly understanding where most data should cluster.
To compute the mean, the formula is:

• \(\mu = \frac{A+B}{2}\)

In our exercise, where \(A = 25\) and \(B = 35\), the mean is \(\mu = \frac{25 + 35}{2} = 30\). This means preparation times are centered around 30 minutes, with an equal spread on either side toward the endpoints, 25 and 35.
Understanding the mean in uniform distribution is critical for identifying scenarios like calculating the likelihood of being within a specific time frame around this average. It shows us that, although each minute is equally probable over the interval, the average represents a balance point of all outcomes.