Question

Explain why the Poisson distribution would be a good choice for the probability distribution of r in the following scenarios:

i. Finding prehistoric artifacts is a common occurrence. It is reasonable to assume the events are dependent.
ii. Finding prehistoric artifacts is a common occurrence. It is reasonable to assume the events are independent.
iii. Finding prehistoric artifacts is a rare occurrence. It is reasonable to assume the events are independent.
iv. Finding prehistoric artifacts is a rare occurrence. It is reasonable to assume the events are dependent.

What is λ? Write out the formula for the probability distribution of the random variable r. (Use e, λ, and r in your answer.) P(r) =

A) λ = 1, P(r) = (e^(-λ) * λ^r) / r!
B) λ = 2, P(r) = (e^(-λ) * λ^r) / r!
C) λ = 3, P(r) = (e^(-λ) * λ^r) / r!
D) λ = 4, P(r) = (e^(-λ) * λ^r) / r!

Answer 1

The Poisson distribution is appropriate for modeling the number of independent prehistoric artifact findings over time, where λ is the mean number of occurrences. This distribution can be appropriate for rare events occurring independently. The probability formula for the Poisson distribution is P(r) = (e^(-λ) * λ^r) / r!.

Step-by-step explanation:

The Poisson distribution is suitable for modeling the number of events occurring in a fixed interval of time or space under certain conditions. Here are explanations for its appropriateness in different scenarios:

• Finding prehistoric artifacts is a common occurrence, and it is reasonable to assume the events are independent: The Poisson distribution is apt because it models events with a known average rate that occurs independently.
• Finding prehistoric artifacts is a rare occurrence, and it is reasonable to assume the events are independent: The Poisson distribution works well for rare events because it can give the probability of these events occurring within a given interval, assuming each event happens independently and at a constant average rate.

The scenarios assuming dependent events are less suitable for Poisson distribution because it assumes independence. However, it might still be used if the dependence is weak.

λ (lambda) is the mean number of occurrences in a given interval. The formula for the probability distribution of the random variable r is given by:

P(r) = (e^(-λ) * λ^r) / r!

This formula represents the probability of having exactly r events occur in a given interval, given a Poisson distribution with mean λ.