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The probability density function (PDF) is a key component when working with the exponential distribution. It tells us how likely it is to encounter a particular time between successive events. For an exponential distribution, the PDF is defined mathematically as: \[ f(x) = \lambda e^{-\lambda x} \] where \(\lambda\) is the rate parameter and \(x\) is the time variable. It is essential to note that this function only applies for non-negative \(x\), meaning \(x \geq 0\). In our specific case, where \(\lambda = 1\), the PDF simplifies to \(f(x) = e^{-x}\). This simplification is due to the standard parameter choice. The role of the PDF is to provide a comprehensive description of the distribution's behavior at each point along its domain. It shows how the probability diminishes exponentially with increasing \(x\). This is a crucial concept to grasp as it lays the foundation for calculating probabilities like \(P(X \leq 4)\). Understanding the PDF lets us visualize how frequently or infrequently certain waiting times occur.

The cumulative distribution function (CDF) helps us find the probability that the random variable \(X\) is less than or equal to a certain value. Unlike the PDF, which gives probabilities for specific time intervals, the CDF accumulates these probabilities up to a point, hence the name "cumulative." For the exponential distribution, the CDF is defined by the formula: \[ F(x) = 1 - e^{-\lambda x} \] In our problem, with \(\lambda = 1\), this simplifies to \(F(x) = 1 - e^{-x}\). By using this expression, we can compute specific probabilities such as \(P(X \leq 4)\). We find that \(P(X \leq 4)\) is approximately 0.9817. This value tells us nearly 98.17% of the time, the next arrival will be within 4 time units. Additionally, to determine probabilities between two points, like \(P(2 \leq X \leq 5)\), we compute the difference \(F(5) - F(2)\), utilizing the properties of the CDF. This exemplifies how cumulative probabilities aid in understanding the likelihood of occurrence within ranges.

The expected value is a fundamental concept in probability and statistics, representing the average or mean value of a random variable. For an exponential distribution, the expected value is mathematically represented as: \[ E(X) = \frac{1}{\lambda} \] With our parameter \(\lambda = 1\), the expected value becomes \(E(X) = 1\). This tells us that, on average, the time between two successive arrivals is 1 unit. The expected value is essential in many practical applications, serving as a measure of the distribution's center. It's akin to finding the balance point of the distribution, providing a single summary statistic that captures the central tendency of the random variable. It's not only used for this specific problem but also as a general tool in probabilistic models and real-world predictions related to time until next events, reliability testing, and more.

The standard deviation measures the dispersion or variability of a set of data points around the mean, giving insights into how spread out the values are. Higher standard deviation indicates more variability. For the exponential distribution, the standard deviation is equal to the expected value and is given by: \[ \text{SD}(X) = \frac{1}{\lambda} \] In our exercise, since \(\lambda = 1\), the standard deviation is also \(1\). This equality highlights a unique characteristic of the exponential distribution: its mean and standard deviation are the same. This metric is important when assessing the reliability and consistency of waiting times between events. It informs us about the variance or uncertainty surrounding the average time calculated in the expected value. By understanding the standard deviation, we gain a deeper comprehension of how often arrival times deviate from the average, which is invaluable in order planning, logistics, and various applications in statistics and engineering.