Poisson Distribution and its Mean

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. It is commonly used to model events that occur randomly and independently over a fixed interval, such as the number of customers arriving at a store or the number of phone calls received in a call center.

The mean of a Poisson distribution, denoted by λ (lambda), represents the average number of events that occur in the given interval. It is a measure of central tendency and characterizes the distribution.

Formula for the Mean of a Poisson Distribution

The mean of a Poisson distribution is given by the formula:

µ = λ

Where:
- µ represents the mean of the distribution
- λ represents the average number of events per interval

Explanation of the Correct Answer (Option B)

The correct answer to the given question is option B: np.

In a Poisson distribution, the parameter λ represents the average number of events per interval. The mean of the distribution is equal to this parameter λ.

To calculate the mean of a Poisson distribution, we can use the formula µ = λ. In this formula, λ represents the average number of events per interval.

In the context of the Poisson distribution, the parameter λ is often defined as the product of the average number of events per unit of time (or space) and the length of the interval. It can be represented as λ = np, where n is the number of units of time (or space) and p is the average number of events per unit of time (or space).

Substituting λ = np into the formula for the mean, we have:

µ = np

Therefore, the mean of a Poisson distribution is equal to np, which corresponds to option B.

Conclusion

In summary, the mean of a Poisson distribution is equal to np, where n represents the number of units of time (or space) and p represents the average number of events per unit of time (or space). This formula allows us to calculate the mean of a Poisson distribution and understand the average number of events occurring in a given interval.