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In a Poisson distribution, the Cumulative Distribution Function (CDF) helps us determine the probability that a random variable is less than or equal to a particular value. For instance, when calculating the probability of having at most 10 drivers, we sum up the probabilities of having 0, 1, 2,..., up to 10 drivers.

This is given by the formula:

• \[ P(X \leq 10) = \sum_{x=0}^{10} \frac{e^{-\lambda} \cdot \lambda^x}{x!} \]

where \( \lambda = 20 \) in our case. By evaluating this summation, we capture all relevant probabilities up to 10. Conversely, to find the probability of having more than 20 drivers, we use the complement rule:

• \[ P(X > 20) = 1 - P(X \leq 20) \]

In essence, the CDF is an essential tool in finding probabilities over specific intervals in a Poisson distribution.

For a Poisson distribution, mean and variance play pivotal roles. They are not just curious metrics—they define the shape of the distribution and how much spread there is in our data.

In the Poisson setup, the parameter \( \lambda \) serves as both the mean and the variance. This is a unique characteristic where:

• Mean \( \mu = \lambda \)
• Variance \( \sigma^2 = \lambda \)

Given \( \lambda = 20 \), both our mean and variance are 20. This makes interpreting the distribution straightforward: we expect around 20 drivers during the time period, with variability also being characterized by the same value. Understanding this helps in setting the stage for further calculations, like standard deviation and probabilities.

The standard deviation provides insights into how much the data deviates from the mean, giving us a sense of spread or dispersion within the dataset. For the Poisson distribution, computing the standard deviation is direct since it is the square root of the variance.

• \( \sigma = \sqrt{\lambda} \)

Given \( \lambda = 20 \), the standard deviation turns out to be approximately 4.47.This measurement allows us to determine how far a typical value lies from the mean. To further utilize this concept, we look at probabilities within a certain range, termed as standard deviation bounds. For instance, being within two standard deviations from the mean implies considering values from \( 20 \pm 2\times4.47 \), which can simplify to 11 to 29 drivers.

Calculating probabilities in the realm of the Poisson distribution can initially seem daunting, but the methods simplify the process.

The methods include:- **Using the Poisson Probability Formula:** This directly computes the probability for a specific number of events (like drivers).

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

- **Using the Cumulative Distribution Function (CDF):** This sums probabilities for a range of values, streamlining the process of finding cumulative probabilities.For our specific cases in the exercise:

• "At most 10 drivers" employs the sum \( \sum_{x=0}^{10} \), aligning our answer with real-world expectations.
• Being "more than 20 drivers" relies on complementarity (i.e., \( 1 - P(X \leq 20) \)).
• To find intervals, like strictly between 11 and 19, we use subtraction within CDF calculations.

These methods ensure precise probability calculations and provide a clear understanding of event likelihoods within a Poisson-distributed scenario.