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The mean in a Poisson distribution is the average number of events (in this case, tornadoes) expected to happen in a fixed interval of time (one year). To calculate the mean, you divide the total number of events by the total number of intervals. Here, we have 153 tornadoes over 64 years, so the mean \( \text{λ} \) is calculated as: \[ \text{λ} = \frac{153}{64} \] Simplifying that, we get \[ \text{λ} = \frac{153}{64} = 2.39 \] This means that, on average, New Mexico experiences approximately 2.39 tornadoes per year. The mean is a central concept in probability distributions as it provides a measure of central tendency.

Understanding the mean will help you predict the frequency of events, which can be crucial for planning and safety measures.

Standard deviation provides a measure of how spread out the values in a distribution are. For a Poisson distribution, the standard deviation is simply the square root of the mean. This is represented mathematically as: \[ \text{σ} = \text{√λ} \] Substituting the mean we calculated earlier (\text{λ} = 2.39), we get: \[ \text{σ} = \text{√2.39} ≈ 1.55 \]

So, the standard deviation is approximately 1.55. This tells you that the number of tornadoes per year typically varies by about 1.55 tornadoes from the average of 2.39. In simpler terms, if you were to gather data over multiple years, most values would fall within this range around the average.

Variance measures how far the numbers in the data set are spread out from their average value. In a Poisson distribution, the variance (\text{σ^2}) is equal to the mean (\text{λ}). So, using our mean of 2.39: \[ \text{σ^2} = \text{λ} = 2.39 \]

Variance provides insight into the different possibilities of outcomes. In this case, a variance of 2.39 implies that while the average number of tornadoes is 2.39, individual years might have somewhat more or fewer tornadoes. This additional layer of understanding can be valuable in predicting outliers and understanding overall behavior.

The Poisson distribution is a probability distribution that measures the likelihood of a given number of events happening within a fixed interval of time. It is especially useful for predicting the number of times an event will occur in a specified period.

For example, the probability of different numbers of tornadoes occurring in a year can be predicted using the Poisson probability formula: \[ P(x; \text{λ}) = \frac{\text{λ}^x e^{-\text{λ}}}{x!} \] where \( \text{λ} = 2.39 \).

This formula helps you find the probability of observing any specific number of tornadoes in a year. For instance, choosing \( x = 3 \) gives the probability of exactly three tornadoes occurring in one year. Understanding this distribution helps in many fields, from meteorology to finance, by providing a mathematical foundation for predicting rare events.