My problem is 7.6 from A First Course in Probability and Statistics. The answer is provided in the book but not how to arrive at the solution. I thought I understood the chapter fairly well, but I can't seen to solve this problem:

"Defects in a particular kind of a metal sheet occur at an average rate of one per 100 square metres. Find the probability that two or more defects occur in a sheet of size 40 square metres."

Answer: 0.0616

I thought the problem could be approached as a binomial distribution where the r.v.s are {0,1} per square meter, and then I could use the Poisson Approximation to the Binomial Distribution to solve the problem (if necessary). This didn't work, and in retrospect, nothing in the problem indicates there can't be more than 1 defect in a square meter. This suggests a continuous distribution rather than a discrete distribution. But only given an expected value and the knowledge that the distribution is probably continuous seems to be too little information. Even when I take a step further an assume a uniform distribution, I still don't see an obvious solution. The associated chapter mainly focuses on Chebyshev's Inequality (which requires the variance), the Weak Law of Large Numbers, and the Central Limit Theorem. None seem applicable given the minimal information.

What am I missing?

  • $\begingroup$ Clearly you have to use the Poisson distribution. Why didn't it work for you? What value of λ did you use? $\endgroup$ – Zahava Kor Jan 14 '18 at 3:05
  • $\begingroup$ I used λ = np = 40 * (1/100) $\endgroup$ – Campbell Jan 14 '18 at 13:48
  • $\begingroup$ λ =0.4 is correct. If you calculate 1 - P(0) - P(1) according to the Poisson distribution with this λ you will get the right answer. $\endgroup$ – Zahava Kor Jan 15 '18 at 2:19

As Zahava Kor pointed out, λ =0.4 is correct. And my initially suspicion of using the Poisson distribution was right after all.

$$ 1 - P(0) - P(1) $$ $$ 1-{e^{-0.4}}\frac{0.4^0}{0!}-{e^{-0.4}}\frac{0.4^1}{1!} $$ $$ 0.0616 $$


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