91Ó°ÊÓ

In statistics, a probability density function (PDF) is crucial for understanding how probability is distributed across a continuous random variable. Think of it as a smooth curve that shows where values are most likely to occur. For a uniform distribution, this function is particularly straightforward.

Since all outcomes are equally likely, the function is constant. In mathematical terms, for a uniform distribution between two points, \(a\) and \(b\), the PDF is given by:

• \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\)

In this exercise, the driving distance is uniformly distributed between 284.7 and 310.6 yards. This means each distance between these two values is equally probable. Therefore, the PDF looks like a flat line on a graph from 284.7 to 310.6.

Once we plug in the given values, we find that \(f(x) = \frac{1}{25.9}\). This implies that any specific distance within this range is equally likely, at a probability density of \(\frac{1}{25.9}\). This doesn’t represent the probability of an exact distance, but rather the density or distribution of probabilities for ranges of distances.

Integration helps us determine the area under the curve of a probability density function over a specific interval. This area corresponds to the probability that the variable falls within that range. It's a key tool in probability as it translates a density function into actual probabilities.

For the uniform distribution of driving distances, integrating the PDF across a range provides the probability that a golfer's drive falls in that range. The formula used in the solution is:

• \(P(x_1 \leq x \leq x_2) = \int_{x_1}^{x_2} \frac{1}{b-a} \, dx\)

This simple integration essentially calculates the width of the range times the constant density \(\frac{1}{25.9}\) established by the PDF. For instance, to find the probability that a golfing drive is less than 290 yards, we integrate from 284.7 to 290 as follows:

\(P(x < 290) = \int_{284.7}^{290} \frac{1}{25.9} \, dx = \frac{5.3}{25.9}\).

This outcome gives the probability of hitting a drive shorter than 290 yards.

Probability calculation in the context of a uniform distribution involves straightforward steps once the PDF is known. Since the uniform distribution treats all intervals within its range equally, calculating probabilities involves determining how much of the total range your interval of interest occupies.

Let's look at some specific probability questions from the exercise:

• Less than 290 yards: Calculate \(P(x < 290)\) by integrating from 284.7 to 290. We find: \(\frac{5.3}{25.9}\), a relatively simple calculation given that the main task is to find the range length 5.3 yards.


• At least 300 yards: Apply similar logic from 300 to the upper limit, 310.6 yards. Thus, \(\frac{10.6}{25.9}\).

The beauty of uniform distribution is that all you need is the interval length over the whole range to find probabilities.

For more complex boundaries like between 290 and 305 yards, it involves the same core steps: \(\frac{15}{25.9}\). It’s highly intuitive and more a matter of simple arithmetic given the uniform nature of the distribution.

The uniform probability distribution is one of the simplest forms of distribution in statistics, defined by its equal-likelihood property. In this distribution, each value within an interval is equally probable.

Think of it like slicing a cake into equal pieces: no piece is bigger or more likely to be picked than any other if you close your eyes and randomly choose.

• It is defined over a specific range (from \(a\) to \(b\)).
• The PDF remains constant across this entire range.

This makes calculations remarkably straightforward. To find how many golfers drive at least 290 yards, we acknowledge that the probability \(P(x \geq 290)\) is the complementary probability to \(P(x < 290): 1 - \frac{5.3}{25.9}\).

Multiplying this result by 100 gives the expected number of golfers achieving this distance in our sample of 100, emphasizing the practical usefulness of the uniform distribution when dealing with limited or equal-interval data.