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The exponential distribution is a continuous probability distribution used to model the time until the next event occurs, such as the time between successive arrivals in a queue. It is characterized by a single parameter, the rate \( \lambda \), which is the average number of events per time interval. The random variable \( X \) representing the time between events follows the exponential distribution, denoted as \( X \sim \text{Exponential}(\lambda) \).
The probability density function (pdf) for an exponential distribution is given by:

• \( f(x) = \lambda e^{-\lambda x} \quad \text{for} \quad x \geq 0 \)

If \( \lambda \) is large, then events happen frequently, meaning shorter time intervals between events. Conversely, a small \( \lambda \) indicates less frequent events. This distribution is widely used in fields like telecommunications, survival analysis, and reliability engineering.

The Gamma distribution generalizes the exponential distribution and is useful for modeling the sum of multiple independent exponential random variables. It has two parameters: the shape parameter \( n \), which corresponds to the number of exponential variables being summed, and the rate parameter \( \lambda \), which is common to the exponential distributions.
The distribution can be denoted by \( X \sim \text{Gamma}(n, \lambda) \). Its probability density function is more complex than that of a single exponential distribution:

• \(f(t) = \frac{\lambda^n t^{n-1} e^{-\lambda t}}{(n-1)!}, \quad t > 0\)

This pdf shows the probability of a total time \( t \) for the completion of all events. When calculating the time at which the last rider departs from the cable car, we use the Gamma distribution to account for the cumulative waiting time from multiple stops.

The memoryless property is a unique characteristic of the exponential distribution. It means that the probability of an event occurring in the next interval is independent of how much time has already elapsed. Formally, if \( X \) is exponentially distributed with rate \( \mu \), this property is expressed as:

• \(P(X > t + s \mid X > t) = P(X > s) = e^{-\mu s}\)

This property simplifies the analysis of random processes, like the time it takes each rider to walk home. Regardless of the time already passed since a rider started walking, the probability of them taking an additional time \( s \) remains unchanged. This feature is beneficial in modeling memoryless processes, simplifying the calculation of event probabilities.

Independent random variables are fundamental in probability theory because they ensure that the occurrence of one event provides no information about the occurrence of another. In this scenario, each of the times for riders to depart and walk home are modeled as independent exponential random variables.
When dealing with independent random variables, particularly when calculating joint probabilities or sums (such as with the Gamma distribution), the independence assumption allows straightforward probability computations. For instance, if the time for each rider's walk home is independent, the overall probability that several riders are home by a certain time is simply the product of their individual probabilities.
This independence is critical when calculating the likelihood that all riders are home by a given time, as shown in the probability expression \( P(Home) = (1 - e^{-\mu t})^{n-1} \) for riders.