91Ó°ÊÓ

Probability calculation is all about determining the likelihood of an event occurring. In this exercise, we dealt with the number of parking tickets issued, and since these occurrences follow a Poisson distribution, we are interested in discrete events over a fixed period. For example, to find the probability of issuing between 35 and 70 tickets in a day, we calculated the probability of the Poisson variable falling within the range of 35 to 70.
This involved using the formula for the Poisson probability mass function. However, as the calculations can become complex, approximations become necessary. When \(\mu\), the mean of the Poisson distribution, is large, it is often useful to approximate it using a normal distribution. This transition helps simplify probability calculations significantly but might not capture precise probabilities, prompting software use for exact results.

Normal approximation is a technique used to simplify the calculation of Poisson probabilities. When the mean (\( \mu \)) is large, as is the case in this exercise, the Poisson distribution is approximately normal. This concept allows us to transform a problem from the realm of Poisson distributions to that of normal distributions, which are more straightforward to work with.
To perform the normal approximation, we calculate the standard deviation using the formula \( \sigma = \sqrt{\mu} \). For instance, with \( \mu = 50 \), the standard deviation becomes approximately 7.07. By converting the range of interest (e.g., 35 to 70 tickets) into a corresponding range of \( Z \) scores, using \( Z = \frac{X - \mu}{\sigma} \), we transform the Poisson problem into a normal probability calculation. Tables or software for the standard normal distribution then help determine the approximate probabilities for the scenarios described.

Standard deviation, denoted by \( \sigma \), is a measure of the amount of variation or dispersion in a set of values. For a Poisson distribution, the standard deviation is particularly easy to calculate using the formula \( \sigma = \sqrt{\mu} \), where \( \mu \) is the average rate of occurrence.
In this exercise, knowing the standard deviation allowed us to perform normal approximations by transforming the original data into standard normal variables. For instance, when calculating the probability of parking tickets in a weekday, the standard deviation of the normal approximation was approximately 7.07, derived from the mean of 50. This value helps convert the problem into the context of normal distribution, facilitating simpler and more familiar probability computations.

Cumulative probability is the probability that a random variable is less than or equal to a certain value. It is an essential concept when dealing with probability distributions like Poisson and normal distributions, as it helps in understanding ranges of interest rather than exact values.
In this exercise, knowing the cumulative probability was crucial for determining the likelihood of events within specific bounds, like finding out how often between 35 and 70 tickets are issued, or how frequently the number of tickets falls between 225 and 275 over a week. By using normal approximation, these probabilities were transformed into cumulative probabilities of the standard normal distribution, which simplified calculations using statistical tables or software tools. This approach offers a holistic view, making it easier to interpret the results and comprehend the probabilities involved.