Final answer:
The question involves the uniform distribution, a type of continuous random variable with a probability density function (PDF) that defines the likelihood of outcomes within a specific range. The area under the PDF represents the probability over an interval, with uniform distribution denoted as X ~ U(a,b). For uniform distribution, probabilities are determined over intervals, not at discrete points.
Step-by-step explanation:
Understanding the Uniform Distribution and Probability Density Function (PDF)
The student's question pertains to the concept of a continuous random variable known as uniform distribution, which is characterized by having equally likely outcomes within a certain range (a, b), and is sometimes referred to as a rectangular distribution because its PDF is shaped like a rectangle. A random variable X with uniform distribution has the notation X ~ U(a,b), and the probability distribution function (PDF) for this continuous random variable is important for calculating probabilities and expressing the likelihood of different outcomes.
In constructing a PDF, one would consider the total area under the curve to be equal to 1, with the probability P(a < x < b) being represented by the area under the PDF curve between points a and b. The mean or average value of a randomly chosen week (or any other period) can be calculated by summing all values in the population and dividing by the number of values. The standard deviation, symbolized by σ, for a uniform distribution is given by the formula σ = √((b-a)²/12).
A key point to understand when dealing with the PDF of a continuous random variable is that the actual value of the function at a point does not represent the probability of the variable taking that exact value, since we are working with a continuous rather than a discrete distribution. Instead, probabilities are determined over intervals. If a problem requires a discrete analogue, such as a geometric distribution, then individual probabilities can be specified directly