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The Poisson distribution is a powerful tool for calculating probabilities, especially when dealing with rare events over a fixed interval. Here, the interval is the number of cars (100), and we are interested in the event of cars exceeding the speed limit.
To calculate probabilities, the Poisson formula is used:

• The formula is: \[ P(x; \lambda) = \frac{\lambda^x e^{-\lambda}}{x!} \]
• Where \(P(x; \lambda)\) is the probability of observing \(x\) events in an interval, \(\lambda\) is the average rate, \(x\) is the specific number of events, and \(e\) is approximately 2.71828.

For instance, to find the probability of exactly 25 cars exceeding the speed limit, you plug in the known values into the formula: \[ P(25; 20) = \frac{20^{25} e^{-20}}{25!} \]Computing this gives the required probability of 25 cars exceeding the limit. Understanding each component of the formula helps in accurately interpreting the questions and solutions related to probabilities in a Poisson distribution scenario.

The cumulative distribution function (CDF) in a Poisson distribution tells us the probability that the variable takes a value less than or equal to a certain point.
This is invaluable when you want to find the probability of observing 'at most' or 'less than' a set number of events.

• For x=0 to x=8, the CDF is the sum of probabilities for all these values.
• The CDF is expressed as: \[ P(X \leq k) = \sum_{x=0}^{k} \frac{\lambda^x e^{-\lambda}}{x!} \]

In practice, the calculation for finding at most 8 cars exceeding the speed involves computing the cumulative probability for each value from 0 to 8. Usually, these probabilities are listed in a Poisson table, which simplifies the process by providing pre-calculated cumulative probabilities for various \(\lambda\) values and \(x\) up to \(k\).
By utilizing CDF, complex cumulative probabilities are made the straightforward sum of accessible tabulated values.

Random sampling is a crucial concept in statistics that ensures the sample observed is representative of the overall population.
In the context of Poisson distribution problems, like the one we are dealing with, random sampling helps in providing accurate probability estimates for events across a sample size of interest.
Understanding how random samples work is important because:

• A random sample maintains the inherent randomness of the population data.
• It eliminates biases, making statistical inferences more powerful.
• It allows the use of statistical distributions, like Poisson, to predict or analyze occurrences within the population based on observed data.

For example, when sampling 100 cars at a school zone, if the cars are randomly chosen, it is more likely that the sample will accurately reflect the broader driving behavior, allowing for trustworthy use of probability calculations.

Lambda (\(\lambda\)) represents the average rate of occurrence for an event within a given interval in a Poisson distribution. Correctly calculating \(\lambda\) ensures accurate probability outcomes.

• The calculation of \(\lambda\) is straightforward: multiply the probability of the event (like cars exceeding the speed limit) by the size of the sample.
• So in our exercise, \(\lambda\) is 20% of 100, given by \( \lambda = 0.20 \times 100 = 20 \).

This parameter is fundamental because it indicates the expected number of occurrences for the event in a given context, providing clarity on the Poisson process's scale.
Accurate \(\lambda\) calculation is crucial for realistic and dependable probability assessments, ensuring that questions involving ranges (like cars at most or at least exceeding speed limits) are correctly approached.