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An exponential distribution describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. In the context of our problem, we use it to model the time between visits to a website.
The key feature of the exponential distribution is its memoryless property. This means the probability of an event occurring in the next unit of time is the same, regardless of how much time has already elapsed.
This makes the exponential distribution useful for modeling situations where we are interested in the time until the next event, like the interval between consecutive website visits.

The probability mass function (PMF) is a function that gives us the probability that a discrete random variable is exactly equal to some value. In the case of a Poisson distribution, the PMF calculates the probability of a given number of events occurring in a fixed interval of time, given the known average rate of arrival, \( \lambda \).
For a Poisson distribution, the formula for the PMF is:

• \( P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \)

Here, \( x \) represents the number of occurrences we are interested in, \( \lambda \) is the mean rate of arrivals per unit time, and \( e \) is approximately 2.71828, the base of the natural logarithm.
Using this function, we can determine the probability of experiencing no visitors within a specified period.

The mean rate of arrival, denoted as \( \lambda \), is a crucial parameter in Poisson and exponential distributions. It represents the average number of events within a given time period. In our example of website visits, \( \lambda = 7 \) indicates that there are, on average, seven visitors per minute.
This rate helps us calculate the expected time between events (visits in this case) and assists in constructing both the exponential density function and the Poisson PMF.
The mean rate directly influences the density and likelihood of occurrences over time, affecting both how quickly events happen and the shape of the distribution curves.

The probability density function (PDF) for an exponential distribution gives us the likelihood of the time before the next event takes place. It is a continuous counterpart to the discrete probability mass function (PMF) used in Poisson distributions.
For an exponential distribution with mean rate \( \lambda \), the PDF is given by:

• \( f(t) = \lambda \cdot e^{-\lambda \cdot t} \)

Here, \( t \) represents the time since the last event, and \( \lambda \) is the constant rate of arrivals that we calculated earlier.
This formula helps calculate the probability of the time until the next event, such as the next website visit, being within a certain range.