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May 9th, 2009, 09:30 PM   #1
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Poisson distribution and binomial distribution questions

Hey i just need to figure out the answers to these questions, not the answers but the method of working, see i dont have a correctr method of working for these questions and i really need them to revise for my upcoming exam. Pl,ease help!!!
Thanks in advance!
1.
(a) Write down the binomial distribution for x successes in n independent trials each with probability p of success.

(b) Write down the Poisson distribution with mean ?.

(c) What is the relation between the Poisson distribution and the binomial distribution?

(d) A factory produces identical items. On average, 0.2% of the items are known to be defective. If a random sample of 500 items is inspected, what is the probability of there being no more than 3 defective items?

2.
(a) Define the Poisson probability distribution with mean ?.

(b) Write down the binomial distribution for x successes in n independent trials each with probability p of success.

(c) On average, 0.15% of the nails manufactured at a factory are known to be defective. If a random sample of 400 nails is inspected, what is the probability of there being no more than 3 defective nails?

3.
(a) Define the Poisson probability distribution with mean p.

(b) A tool hire shop has six lawn mowers which it hires out on a daily basis.The number of lawn mowers requested per day follows a Poisson probability distribution with mean 4.5. Find the probability that:

(i) exactly three lawn mowers are hired out on any one day;
(ii) all lawn mowers are in use on any one day.
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May 11th, 2009, 06:49 AM   #2
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Re: Poisson distribution and binomial distribution questions

binomial distribution is

P(x) = C(n,x) p^x [(1-p)^(n-x)] , x = 0,1,2, ...n

poisson distribution is

P(x) = e^(-?) (?^x) / (x!) , x = 0,1,2, ....


see the following page for the pdf


http://keral2008.blogspot.com/2008/1...riance-of.html

http://rbmix.com/problem/math/poisson/poisson.html


If X follows B(n,p) and
if n is large and p is small with ? =np (finite),
you can assume that
X follows Poisson distribution with mean ?.

As far as the last part of the first question is concerned
if X is the number of defectives out of 500
and you assume binomial with n=500 , p = 0.002 ( from 0.2%)
or you can assume poisson approximation with ? =np = 500*0.002 =1

substitute into the poisson pdf
and evaluate P(X <=3) = P(X=0)+P(X=1)+P(X=2)+P(X=3)


2(c) is similar

in 3(b) mean =4.5 implies ? =4.5

you are asked to find P(X=3) in part(i)
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