91Ó°ÊÓ

When dealing with continuous random variables, the probability density function (pdf) is an essential concept in probability and statistics. The pdf describes how the probabilities are distributed over the values of the random variable. For any value \( y \), the pdf, \( f(y) \), specifies the relative likelihood of \( Y \) being close to \( y \).

In uniform distribution, the pdf is often flat (constant) over the interval, but this example shows a case with a piecewise linear pdf. For the total waiting time \( Y \), the pdf is defined over different intervals, explicitly outlining how probable each waiting time is.

- For \( 0 \leq y < 5 \), the pdf is an increasing linear function: \( f(y) = \frac{1}{25} y \). This indicates that as \( y \) approaches 5, the likelihood of waiting time \( y \) increases.
- For \( 5 \leq y \leq 10 \), the pdf is a decreasing linear function: \( f(y) = \frac{2}{5} - \frac{1}{25} y \). Here, the likelihood starts high at \( y=5 \) but declines as \( y \) approaches 10.
- Outside of the range \([0, 10]\), the pdf is zero, meaning there's no chance of waiting time being less than 0 or more than 10 minutes.

Visualizing the pdf can help in understanding and calculating probabilities related to the waiting times. Each segment of this function tells us how the waiting times are distributed across different intervals.

Integration is a critical operation in probability, especially when working with continuous probability distributions. For a probability density function, integrating over an interval gives the cumulative probability that the random variable falls within that range.

- The total probability across all possible outcomes must add up to 1. This is expressed mathematically as \( \int_{-\infty}^{\infty} f(y) \, dy = 1 \). This guarantees that the entire area under the pdf curve is equal to 1, ensuring the function is a valid probability distribution.

In practical terms, when we integrate the pdf of the total waiting time \( Y \) over certain limits, we determine the probability that \( Y \) falls within those bounds.

For this exercise:
- To verify that the pdf is valid, check if \( \int_{0}^{10} f(y) \, dy = 1 \). This involves partitioning the integration over the two different linear expressions in the pdf: from 0 to 5 and 5 to 10.

Each integration corresponds to a specific segment of the distribution, and when combined, they confirm the completeness of the distribution. Such integrations are fundamental for calculating any specific probabilities, such as those asking for time windows or specific conditions.

Cumulative probability reflects the total probability that a random variable is less than or equal to a certain value. In mathematical terms, the cumulative distribution function (CDF) quantifies \( P(Y \leq a) \), where \( a \) is a specific waiting time.

In this context, cumulative probability is calculated by integrating the pdf up to the desired point.
Here are examples of how to derive specific cumulative probabilities for given time intervals:

- **Probability for Waiting Time \(\leq 3 \) Minutes:** Calculate \( \int_{0}^{3} \frac{1}{25} y \, dy \). This will yield the likelihood that \( Y \) is 3 minutes or less.
- **Probability for Waiting Time \(\leq 8 \) Minutes:** Compute \( \int_{0}^{5} \frac{1}{25} y \, dy + \int_{5}^{8} \left(\frac{2}{5} - \frac{1}{25} y\right) \, dy \). This tells you the chance \( Y \) is at most 8 minutes.

The integration limits dictate the span over which we are interested. By calculating these integrals, we aggregate probabilities incrementally, leading to a comprehensive understanding of potential outcomes based on the waiting time.