91Ó°ÊÓ

The probability density function (PDF) provides a way of representing the likelihood of all possible outcomes of a continuous random variable. In the context of the triangular distribution in this exercise, the PDF is split into two linear segments. For the interval \(30 < x < 50\), the function is given by \(f(x) = 0.025x - 0.0375\). This equation describes how the probability function rises and reaches the peak of the triangle at the point where \(x = 50\). Here, the triangle has its maximum height.

For the interval \(50 < x < 70\), the function changes to \(f(x) = -0.025x + 0.0875\), which represents the descending linear segment until it approaches zero as \(x\) increases towards 70. The triangular shape of this distribution is created because it rises linearly to a peak, then declines linearly, thus forming a triangle when plotted. It's important to understand that the area under the PDF over its entire range must be equal to 1, as this represents the total probability of all possible outcomes.

Integration in probability refers to the technique of finding the area under the probability density function over an interval. This area represents the probability of the random variable falling within that interval. It is a key tool for calculating probabilities from a PDF.

In this exercise, we use integration to find specific probability intervals. For instance, to find \(P(X \leq 40)\), we integrate the PDF from 30 to 40. The integration of \(f(x) = 0.025x - 0.0375\) over this range gives us the probability that \(X\) is 40 or less. This is calculated by evaluating the definite integral:

• \(\int_{30}^{40} (0.025x - 0.0375) \, dx \)

The area provides a probability of 0.25, indicating a 25% chance that \(X\) is less than or equal to 40.

Similarly, for \(40 < X \leq 60\), integration helps calculate the probability by splitting it into two parts, aligning with the two separate PDF segments in the triangular distribution.

The cumulative distribution function (CDF) provides another way to express probabilities. It's a function that maps every possible value \(x\) of a random variable to the probability that the variable takes a value less than or equal to \(x\).

In a triangular distribution like the one in our exercise, the CDF is derived from integrating the PDF. Each value \(x\) on the CDF represents the accumulated probability from the start of the distribution up to \(x\). This makes the CDF an increasing function, reaching 1 at the endpoint of the distribution range.

The CDF is especially useful for finding the probability that a random variable exceeds a particular value. For instance, when asked to find the value of \(x\) exceeded with a probability of 0.99, we look for the value where the CDF equals 0.01, because the total area under the PDF must equal 1. Thus we deduce that the CDF rapidly approaches the total as we move rightwards across the values of \(x\), helping to identify the point where the given probability is exceeded.