91Ó°ÊÓ

A cumulative distribution function (CDF) represents the probability that a random variable takes on a value less than or equal to a specified number. For the exponential distribution provided in this exercise, the CDF is calculated from the probability density function (PDF). It involves integrating the PDF.

The PDF given is \( f(x) = 0.1 \exp(-0.1x) \) for \( x > 0 \). To get the CDF, \( F(x) \), integrate the PDF from 0 to \( x \):
\[ F(x) = \int_{0}^{x} 0.1 \exp(-0.1t) \, dt = 1 - \exp(-0.1x) \].

The CDF helps in finding the probability of arriving between specific times by subtracting two CDF values, as demonstrated in the problem. If you're asked to find the probability for time intervals, it's handy to use CDF since you only need the values of \( F(x) \) at the end points and then subtract them.

• \( F(15) \) for 8:15 AM is \( 1 - \exp(-1.5) \)
• \( F(30) \) for 8:30 AM is \( 1 - \exp(-3) \)

Thus, \( P(15 < x < 30) = F(30) - F(15) \) gives the probability that you arrive between these two times. This method is particularly efficient and showcases the utility of CDF.

The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials. Here, our interest lies in arriving before 8:40 AM on two or more days out of five.

Each day is considered a trial, and arriving before 8:40 AM is a success. We know the probability of arriving before 8:40 AM on any given day is approximately 0.9817. To compute the probability of achieving this success on two or more days, we make use of the binomial distribution:

For a binomial distribution \( B(n, p) \), where \( n \) is the number of trials and \( p \) is the probability of success, the probability of \( X \) successes can be calculated as:
\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]

In this scenario, \( n = 5 \) (days) and \( p = 0.9817 \). We calculate \( P(X \geq 2) \) by finding:

• \( P(X=0) \)
• \( P(X=1) \)

And then subtracting from 1:
\[ P(X \geq 2) = 1 - P(X=0) - P(X=1) \]
Using the binomial distribution formula makes it easy to find the probability across multiple trials.

An exponential distribution is commonly used to model the time until an event occurs, given its memorylessness property. In this exercise, it models the time you arrive after 8:00 AM.

The PDF of the arrival time is given by:
\( f(x) = 0.1 \exp(-0.1x) \) for \( x > 0 \).
This function reflects the likelihood of arriving at a precise time.

The rate parameter here is \( \lambda = 0.1 \), which indicates how rapidly the event occurs over time. A key trait of the exponential distribution is its simplicity in parameterizing real-world wait times, making it applicable in contexts like service networks and natural processes.

The exponential distribution's memoryless property implies that the future event doesn't depend on how much time has already elapsed, which uniquely suits situations where the timing of an event is purely random. Integrating the PDF over a specified interval gives the probability, which provides practical insights into time-dependent scenarios.

Integral calculus is central for solving problems involving continuous probability distributions like the one given here. It is used to determine the probabilities of certain events from the PDF.

For example, to find the probability that you arrive by a specific time, you need to determine the definite integral of the PDF over a specified range.

The step-by-step solution provided:

• \( \int_{0}^{60} 0.1 \exp(-0.1x) \, dx \) solves for the probability of arriving by 9:00 AM.
• Similarly, \( \int_{15}^{30} 0.1 \exp(-0.1x) \, dx \) captures the probability of arriving between 8:15 and 8:30 AM.
• The integral \( \int_{0}^{40} 0.1 \exp(-0.1x) \, dx \) is used to find the probability of each day before 8:40 AM.

This use of calculus helps in deriving the CDF from the PDF and obtaining specific probabilities through integration. The process is crucial in deriving cumulative probability, explaining why integral calculus empowers solutions in continuous random variable contexts.