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The first step in understanding the Poisson distribution for this problem is to calculate the mean and standard deviation, which are fundamental statistical measures.
The mean, denoted by \( \lambda \), represents the average rate at which events occur.
For a Poisson process where cabs pass by at a rate of 5 cabs per hour, determining the mean for a longer period, like a 10-hour day, involves multiplying the hourly mean by the number of hours: \( \lambda = 5 \times 10 = 50 \) cabs.

• **Mean**, \( \lambda = 50 \) cabs for a 10-hour day.

The standard deviation gives us an idea of the dispersion or spread of the data around the mean. For a Poisson distribution, the standard deviation is the square root of the mean.
Therefore, for our mean of 50 cabs, the standard deviation is \( \sqrt{50} \approx 7.07 \) cabs.

• **Standard Deviation**, \( \sqrt{50} \approx 7.07 \).

When working with Poisson distributions, especially with a large mean, an effective technique is the normal approximation.
This approach simplifies calculations by using the normal distribution as a proxy.
For the problem at hand, we approximate the Poisson distribution with a normal distribution with a mean (\( \mu \)) of 50 cabs and a standard deviation (\( \sigma \)) of approximately 7.07 cabs.

• This allows us to apply continuous distribution methods to estimate probabilities.
• It's essential especially when calculating probabilities that are not straightforward.

This approximation becomes more accurate as the mean of the Poisson distribution increases, due to the Central Limit Theorem which assures that with a large enough mean, the discrete Poisson distribution begins to resemble a continuous normal distribution more closely.

Z-score is a measure of how many standard deviations an element is from the mean.
It is a critical concept when utilizing normal approximation to find probabilities.
For instance, calculating the probability of more than a certain number of cabs requires finding the Z-score for that number.
Using the formula \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the number of cabs, you can compute the Z-score.

• For example, to find the Z-score for 65 cabs: \( Z = \frac{65 - 50}{7.07} \approx 2.12 \).
• This Z-score tells us that 65 cabs are 2.12 standard deviations above the mean number of cabs per 10 hours.

This score is then used to lookup probabilities in the standard normal distribution table, enabling further probability analysis.

Once the Z-score is calculated, the next step is estimating the probability from the standard normal distribution.
This involves finding the area under the normal curve related to the Z-score.
The Z-score can be used to find cumulative probabilities from standard normal distribution tables (Z-tables).

• For a Z-score of 2.12, using Z-tables, \( P(Z > 2.12) \approx 0.017 \) or 1.7%.
• This probability estimates that there is a 1.7% chance more than 65 cabs will pass in a 10-hour period.

Additionally, estimating probabilities for a range, like between 50 and 65 cabs, involves subtracting cumulative probabilities found for each Z-score: \( P(0 < Z < 2.12) = P(Z < 2.12) - P(Z < 0) \).
This practical method leverages the Z-table data to deduce probabilities effectively and efficiently.