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The Poisson probability function is a way to determine the likelihood of a certain number of events happening within a fixed interval of time or space. It's a popular tool when each event occurs independently, and we know the average number of events (called the rate) that occur per unit period.
The function is expressed mathematically as follows: - For a given number of occurrences \( x \), and a known average rate \( \lambda \), the probability \( P(X = x) \) is given by: \[ P(X = x) = \frac{{e^{-\lambda} \lambda^x}}{{x!}} \]

• \( e \) is the base of the natural logarithm, approximately 2.71828.
• \( x! \) ("x factorial") is the product of all positive integers up to \( x \).

Simply put, this formula calculates how likely it is to see exactly \( x \) events when we expect \( \lambda \) events on average.

In probability theory, the expected value is the average of all possible outcomes of a random variable, weighted by their probabilities. For a Poisson distribution, it's quite intuitive.
The expected value is simply the average number of occurrences \( \lambda \) that you expect to see within a specific period of time. So, if you know your rate is 2 occurrences per period, your expected value \( E[X] \) over one period is \( 2 \).
When considering multiple periods, you adjust the expected value accordingly. Thus, for 3 time periods: \[ \lambda_{total} = \lambda \times t = 2 \times 3 = 6 \text{ occurrences} \]

• This reflects the total expected occurrences over the extended timeframe.
• It's a straightforward calculation but highlights how accumulative expected values help in predicting outcomes over extended periods.

Time period analysis in a Poisson distribution involves understanding how the rate of occurrences can change depending on how many time periods are being considered.
In our context, when the rate per time period \( \lambda \) is 2, analyzing three time periods becomes crucial when one needs to predict probabilities for varying occurrences:

• For probability functions across multiple time periods, multiply the \( \lambda \) by the number of periods, thus converting it into \( \lambda_{total} \).
• This total rate reflects the accumulated expected occurrences over the considered timeframe.

So, if you need to calculate the probability for, say, 6 occurrences over 3 periods, use \( \lambda_{total} = 6 \) in the Poisson formula: \[ P(X = x) = \frac{{e^{-6} 6^x}}{{x!}} \]
This expansion of the time frame is essential because it adjusts the probability calculations to consider the longer duration, which could otherwise skew the findings if treated as single-period events.