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The probability distribution function (PDF) for an exponential distribution tells us the likelihood of different time intervals occurring between events. In the case of vehicle arrivals at an intersection, it describes the chances of a particular gap between one car arriving and the next. The exponential distribution is characterized by being right-skewed, which means it has a longer tail on the right. This is typical for distributions that model time between events, as there is no upper limit to how long you might wait for something to happen.

Mathematically, the PDF of the exponential distribution is expressed as: \( f(x; \lambda) = \lambda e^{-\lambda x} \) for \( x \geq 0 \). Here, \( \lambda \) is the rate parameter, which is the reciprocal of the mean.

For a distribution with a mean of 12 seconds, \( \lambda = \frac{1}{12} \). This function starts at a maximum when \( x \) is zero and decreases exponentially as \( x \) increases, never quite reaching zero. This behavior matches situations where very short intervals are common, but longer intervals are possible although less likely.

The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a certain value. In simpler terms, it tells us how likely it is that the time between events will fall within a given range. For example, it could be used to find the probability that the time between car arrivals is 12 seconds or less.

For the exponential distribution, the CDF is given by:\( F(x; \lambda) = 1 - e^{-\lambda x} \) for \( x \geq 0 \).This formula helps calculate the total probability of events occurring up to a certain point \( x \).

Using \( \lambda = \frac{1}{12} \), if you want to find out the probability of 12 seconds or less between arrivals, substitute \( x = 12 \) into the CDF: \( P(X \leq 12) = 1 - e^{-1} \approx 0.6321 \). This result shows that there's a 63.21% chance that the arrival time will be 12 seconds or less.

Similarly, for 6 seconds, the calculation \( P(X \leq 6) = 1 - e^{-0.5} \approx 0.3935 \) indicates a 39.35% probability for the time interval being 6 seconds or less.

The mean of a distribution is a measure of central tendency, indicating where the center of the data tends to be. For an exponential distribution, the mean is the average time between events, such as vehicle arrivals. It's an essential parameter because it helps define the shape and scale of the distribution.

In the exponential distribution, the mean \( \mu \) is the reciprocal of the rate \( \lambda \), expressed as \( \mu = \frac{1}{\lambda} \). For our problem, where the time between vehicle arrivals follows an exponential distribution with a mean of 12 seconds, \( \lambda = \frac{1}{12} \).

This measure is crucial because it summarizes the typical waiting time between events. While knowing the mean, you can understand how frequently events are expected to occur. In real-life applications, this knowledge helps in planning and optimizing operations, ensuring efficiency in managing processes like traffic flow at intersections.