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The expected value, often referred to as the mean, is a fundamental aspect of probability and statistics. In the context of the exponential distribution, it helps us understand the average time between events. For an exponential distribution with a rate parameter \( \lambda \), the expected value is given by:\[\frac{1}{\lambda}.\]This formula simplifies the process of finding the mean by inverting the rate parameter. For our exercise, with \( \lambda = 1 \), the expected time between successive arrivals at the bank's drive-up window is simply \( 1 \) unit of time. This means that, on average, one can expect a minute or similar measure of time between two arrivals based on this distribution.

Standard deviation is a measure of the amount of variation or dispersion of a set of values. For the exponential distribution, which deals with the time between events in a process that happens continuously and independently at a constant average rate, the standard deviation is a vital metric.In the exponential distribution, the standard deviation is the same as the expected value, given by:\[\frac{1}{\lambda}.\]As with the expected value, if \( \lambda = 1 \), the standard deviation is also \( 1 \). This highlights the uniform nature of the exponential distribution where the mean and standard deviation align, reflecting the consistency of the time intervals between events in a predictable manner.

The cumulative distribution function (CDF) of a probability distribution describes the probability that a random variable takes a value less than or equal to a specific value. For the exponential distribution, the CDF is represented as:\[F(x; \lambda) = 1 - e^{-\lambda x}.\]This formula calculates the probability that the time between events is at most \( x \). It inversely describes the exponential decay of time between events. For example, in our problem, the CDF for \( X \leq 4 \) is computed by substituting \( x = 4 \) and \( \lambda = 1 \), giving:\[P(X \leq 4) = 1 - e^{-4} \approx 0.9817.\]This indicates a high likelihood that the time between arrivals is four units or less, illustrating the capability of the CDF to quantify time-related probabilities.

Calculating probabilities for specific ranges is a common task when dealing with exponential distributions. It allows us to determine the likelihood of an event occurring within a defined period. Probabilities in certain ranges are obtained by finding the difference between two CDF values.For example, to find \( P(2 \leq X \leq 5) \), compute the probabilities for both bounds:

• \( P(X \leq 5) = 1 - e^{-5} \approx 0.9933 \)
• \( P(X < 2) = 1 - e^{-2} \approx 0.8647 \)

Subtracting these gives the probability for the interval:\[P(2 \leq X \leq 5) = 0.9933 - 0.8647 \approx 0.1286.\]This tells us the likelihood of the event occurring within this specific time frame.