91Ó°ÊÓ

In statistics, a Probability Density Function (PDF) explains how the range of possible values of a continuous random variable is distributed. For a uniform distribution, the PDF is constant between two boundary points, denoted as \(a\) and \(b\), and zero otherwise. This means every value between \(a\) and \(b\) is equally likely. The function creates a rectangle shape when graphed over the interval.

For a uniform distribution over \([a, b]\), the height \(h\) of the rectangle (PDF) is calculated as \( \frac{1}{b-a} \). In the exercise, with \(a=2\) and \(b=5\), the probability density is \( \frac{1}{3} \).

The most important property of a PDF is that the area under its curve equals 1, which is the entire probability of the defined interval. Essentially, this means that the sum of all probabilities for continuous data between \(a\) and \(b\) should equal 1. In our case, the total area under the curve is \(h \times (b-a) = \frac{1}{3} \times 3 = 1\).

The mean and standard deviation are crucial concepts for understanding the characteristics of a distribution.

**Mean of a Uniform Distribution:**

• The mean, or average, represents the central point of the distribution and is symbolized as \(\mu\).
• For a uniform distribution, the formula for the mean is \( \mu = \frac{a+b}{2} \).
• In our problem, substituting \(a=2\) and \(b=5\), the mean is \( \mu = \frac{2+5}{2} = 3.5 \).

**Standard Deviation of a Uniform Distribution:**

• The standard deviation represents the spread or variability of the data around the mean.
• In a uniform distribution, it's derived as \( \sigma = \frac{b-a}{\sqrt{12}} \).
• For the given \(a\) and \(b\) values, the standard deviation is \( \frac{3}{\sqrt{12}} \approx 0.866 \).

When dealing with continuous data, a continuous probability distribution is necessary to describe the probability of outcomes over an interval. Unlike discrete distributions, which consider distinct values, continuous distributions analyze any value within a specified range.

Uniform distribution is a prime example of a continuous probability distribution. It signifies that each point within a given interval is equally probable, leading to a horizontal-line-shaped PDF over that range.

In practice, a continuous probability uniform distribution between 2 and 5 (our example interval) implies that a value picked randomly from this range has a consistent probability of occurrence. Thus, the probabilities for any subset of this range can be calculated by pinpointing the interval's length and multiplying it by the height (which is the constant probability density), like seen in steps 5 and 6 of the solution.