91Ó°ÊÓ

A probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. Unlike discrete variables, continuous variables can take any value within a given range. In the given exercise, the density function is defined as \( f(x)=\frac{1}{8}x, \; 0 \leq x \leq 4 \).
This linear function increases from 0 to 4, which means as \( x \) gets larger, the value of the PDF increases. The purpose of the PDF is to assign probabilities to different ranges or intervals within the variable’s domain.
It's important to note that the PDF itself does not give probabilities directly; rather, it highlights the density of probability across a range of values. The total area under the PDF curve over the entire range must be 1, representing a 100% probability.

Integration is a fundamental concept in calculus used to calculate the area under a curve. When dealing with a continuous random variable, we use integration to find the probability of the variable falling within a certain range. In the context of the given exercise, we calculate probabilities by integrating the PDF over the specified ranges.
For instance, to find \( \operatorname{Pr}(X \leq 1) \), we integrate the function \( \frac{1}{8}x \) from 0 to 1.

• Set up the integral: \( \operatorname{Pr}(X \leq 1) = \int_{0}^{1} \frac{1}{8} x \; dx \)
• Evaluate the integral: \( \int_{0}^{1} \frac{1}{8} x \; dx = \left[ \frac{1}{16}x^2 \right]_0^1 = \frac{1}{16} \)

This process is repeated for other intervals to determine different probabilities within the given function's range.

To calculate the probability for different intervals of a continuous random variable, we follow the process of integration. In the exercise, we calculated three specific probabilities:

• \( \operatorname{Pr}(X \leq 1) \): By integrating \( \frac{1}{8} x \) from 0 to 1, we get \( \frac{1}{16} \).
• \( \operatorname{Pr}(2 \leq X \leq 2.5) \): By integrating \( \frac{1}{8} x \) from 2 to 2.5, we obtain \( \frac{9}{64} \).
• \( \operatorname{Pr}(3.5 \leq X) \): By integrating \( \frac{1}{8} x \) from 3.5 to 4, we get \( \frac{15}{64} \).

Each of these steps involves setting up and evaluating a definite integral, which calculates the area under the curve between the specified limits. The resulting value gives us the probability for that interval.

Graphing functions is an important aspect of understanding the behavior of continuous random variables. In the exercise, we graph the given density function \( f(x) = \frac{1}{8}x, \; 0 \leq x \leq 4 \).
Steps to graph the function include:

• Identify key points: At \( x = 0 \), \( f(0) = 0 \). At \( x = 4 \), \( f(4) = \frac{1}{2} \).
• Draw the curve: The function is linear, so we draw a straight line connecting the points (0,0) and (4, 0.5).

This graph shows an increasing trend, indicating that the probability density increases with \( x \). Once the graph is drawn, we can shade specific areas under the curve to visually represent the probabilities for different intervals. For example, shading the area from 0 to 1 helps visualize \( \operatorname{Pr}(X \leq 1) \). Graphing aids in understanding how the function behaves and how probabilities are distributed over the range.