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Probability, in simple terms, is a measure of how likely an event is to occur. In the context of Poisson distribution, which is used to model the probability of a given number of events happening in a fixed interval of time or space, these events must happen independently and with a known constant mean rate.
In our shoplifting example, we're interested in the probability of shoplifting incidents over the 11-hour period during which the shopping center is open. The Poisson distribution provides a straightforward way to calculate such probabilities when incidents occur rarely or are spread out over time.
This distribution uses the rate of occurrences (\(\lambda\),) to calculate specific probabilities. For example, if we want to find out the likelihood of exactly \(r\) incidents within a given time frame, we use the Poisson probability mass function:\[ P(r) = \frac{e^{-\lambda} \cdot \lambda^r}{r!} \]where:

• \(e\) is Euler's number (approximately 2.71828),
• \(\lambda\) is the average rate of occurrence,
• \(r\) is the number of incidents we are interested in.

In the Poisson distribution, \(\lambda\) is a crucial parameter representing the average number of occurrences within the specified interval. It can be thought of as the mean of the distribution.
For our shopping center scenario, \(\lambda\) is calculated based on the average shoplifting rate. With an incident due every 3 hours and the store being open for 11 hours, we determine \(\lambda\)as:\[ \lambda = \frac{11}{3} \approx 3.67 \]This means we expect around 3.67 incidents during the open hours.
The ability to calculate and understand \(\lambda\)is fundamental when using the Poisson distribution, as it directly influences the probability calculations for varying numbers of incidents.

The random variable \(r\) in our scenario denotes the number of shoplifting incidents observed in the 11-hour period while the shopping center is open.
In the realm of probability distributions, a random variable is a variable that can take various values based on the outcomes of a random phenomenon. Here,\(r\)can take values like 0, 1, 2, with each representing different numbers of shoplifting incidents.
The concept of a random variable is foundational when dealing with statistical distributions, as it helps translate real-world random events into quantifiable variables that can be systematically analyzed. In Poisson distribution, the probabilities associated with different \(r\)values are calculated typically using the formula mentioned previously, and they allow us to draw insights about how common or rare different numbers of incidents are. This is crucial for understanding not just theoretical probabilities, but also for making informed predictions and decisions in real-world scenarios.