Question

Formulate the Poisson-governed housing inventory damage problem from class, and estimate the Probability of Damage P(D) using "damage" as the events (as opposed to "undamaged" as done in class). Then, determine P(D) exactly for various levels of p=[0.01,0.05,0.10,0.20]. Discuss results.

Answer 1

The student's question on housing inventory damage can be addressed using a Poisson distribution, where p represents the probability of damage. The calculation for P(D) involves using the Poisson formula for different values of p and noting the trend as p changes.

Step-by-step explanation:

To address the problem of housing inventory damage using a Poisson distribution, we assume that 'damage' is a rare event occurring with a known average rate in a large population of houses. When estimating the Probability of Damage P(D), we can use the Poisson formula: P(X=k) = (e-λλk)/k!, where λ is the average rate of damage (the product of the number of trials n and the probability of a single damage event p), and k is the number of damage events we're interested in finding the probability for (in this case, we are interested in at least one damage event so k ≥ 1).

For different levels of probability p, keeping in mind the number of trials n is large, we can estimate P(D) as follows:

1. For p = 0.01, find λ by multiplying n and p, then use it to calculate P(D).
2. Repeat this process with p = 0.05, 0.10, and 0.20.
3. Discuss the trend observed as p increases; typically, as p increases, P(D) will increase up to a point before it starts decreasing again, due to the nature of the Poisson distribution.

Remember, the approximation of the binomial distribution with a Poisson distribution is valid when n is large and p is small.