91Ó°ÊÓ

Probability theory is a branch of mathematics that deals with events' likelihoods. When considering the likelihood of daily accidents in a city, probability theory allows us to mathematically predict how often events like accidents might occur. The Poisson Distribution is a tool within this theory specifically designed for predicting the number of events occurring in a fixed interval of time or space when these events occur with a known, constant mean rate and are independent of each other.

In our exercise, knowing there are, on average, 8 accidents daily allows us to use the Poisson probability formula to find various probabilities, such as the chance that no accidents will occur or the possible distribution of accident numbers per day.

The mean, often symbolized by \( \overline{x} \) or \( \mu \), represents the average value of a set of numbers in a probability distribution. In the context of the Poisson distribution, the mean is particularly important as it is equal to the rate \( \lambda \), which is the average number of times an event occurs in a fixed interval. For example, in the given problem, the mean number of accidents is 8. This tells us that if you pick any average day, you'd expect about 8 accidents to occur.

Variance, denoted by \( \sigma^2 \), represents how much the individual numbers in a set differ from the mean. In a Poisson distribution, variance is also equal to \( \lambda \). This indicates that our data's spread, or how much values deviate from the mean, is 8. Therefore, both the mean and the variance in our example are the same: 8.

Standard deviation, denoted by \( \sigma \), is a critical concept in statistics and probability that measures the amount of variation or dispersion in a set of values. In simple terms, it tells you how spread out the numbers are around the mean.

In a Poisson distribution, the standard deviation is calculated as the square root of the variance. Since our problem's variance is 8, the standard deviation is \( \sqrt{8} \), which is approximately 2.83.

This means on any given day, the number of accidents will typically vary by about 2.83 around the average of 8. In essence, it provides a sense of how narrow or wide our range of daily accidents can be.

In probability theory, a probability distribution illustrates how the probabilities are distributed over the values of the random variable. For our scenario with accidents, the random variable \( x \) represents the number of accidents occurring per day, which can range from 0 to potentially any number of events.

Using the Poisson probability formula, we can compute the probability for each possible number of accidents. For instance, the probability of experiencing no accidents (\( x = 0 \)) is determined using the equation: \( P(0; 8) = \frac{8^0 e^{-8}}{0!} \). Likewise, we can calculate probabilities for 1, 2, 3 accidents per day, and so on. By doing this, we form a complete probability distribution showing the likely occurrence of each number of accidents, helping us visualize and understand the data more effectively.