91Ó°ÊÓ

The probability density function (PDF) is a key concept used to describe the behavior of continuous random variables. It provides a powerful tool to understand how probabilities are distributed over different outcomes. For the exponential distribution, the PDF is given by the formula:
\[ f(x; \lambda) = \lambda e^{-\lambda x} \]where:

• \( x \) is the value of the continuous random variable.
• \( \lambda \) is the rate parameter, which is the inverse of the mean \( \mu \) (i.e., \( \lambda = \frac{1}{\mu} \)).

In the context of travel time to work for New York City residents, the average travel time is given as 38.3 minutes. Hence, the rate parameter is \( \lambda = \frac{1}{38.3} \approx 0.0261 \). Using this rate, the PDF describing the travel time for New York City residents is:\[ f(x; 0.0261) = 0.0261 e^{-0.0261 x} \]This function helps us determine the probabilities associated with various ranges of travel times.

Travel time statistics give us insights into how long journeys typically take for commuters, providing a useful basis for calculating probabilities. In our scenario, New York City residents experience an average travel time of 38.3 minutes to get to work.
The exponential distribution is particularly useful for modeling travel times because it represents the time between events in a Poisson process. Assumptions underlying this model include:

• Events occur continuously and independently at a constant average rate over time.
• The probability of more than one event occurring in an infinitesimally small time interval is negligible.

For New York City residents, modeling travel times using the exponential distribution simplifies the probability calculations, leveraging the memoryless property of the exponential function, where the future probability is independent of the past.

Probabilities for specific travel time intervals can be calculated using the PDF and integration techniques. Let's address two cases for New York City residents' travel times:
Probability of Travel Time Between 20 and 40 Minutes
To calculate this, we find the area under the PDF for \( 20 \leq x \leq 40 \):\[ P(20 \leq x \leq 40) = \int_{20}^{40} 0.0261 e^{-0.0261 x} \, dx \]Using calculus, this integral evaluates to:\[ -e^{-0.0261 \times 40} + e^{-0.0261 \times 20} \]Calculations show \( P(20 \leq x \leq 40) \approx 0.2413 \).
Probability of Travel Time More Than One Hour
To find the probability that a commute takes longer than 60 minutes:\[ P(x > 60) = 1 - \int_{0}^{60} 0.0261e^{-0.0261x} \, dx \]This simplifies to:\[ P(x > 60) = e^{-0.0261 \times 60} \approx 0.2096 \]These calculations allow commuters to understand the likelihood of their journey times and plan accordingly.