Poisson Distribution question

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August 31st 2011, 12:21 AM
Paymemoney

Poisson Distribution question

Hi
Can someone tell me if my answers are correct?
A machine makes a special type of lining brick at the rate of 25 per hour. Overall about 5% of the bricks are
defective. Calculate:
The probability that no defective bricks are produced in any one hour?
Now this is a poisson distributions so i used the calculated by using the calculator function
poissonpdf(1.25,0) = 0.2865
however answer is 0.2774

the probability that two defective bricks are produced in any one hour?
poissonpdf(1.25,2) = 0.2238
however answer is 0.2305

the probability that at least one defective brick is produced in the next hour?
not sure about this one

P.S
August 31st 2011, 01:36 PM
chisigma

Re: Poisson Distribution question

Quote:

Originally Posted by Paymemoney

Hi
Can someone tell me if my answers are correct?
A machine makes a special type of lining brick at the rate of 25 per hour. Overall about 5% of the bricks are
defective. Calculate:
The probability that no defective bricks are produced in any one hour?
Now this is a poisson distributions so i used the calculated by using the calculator function
poissonpdf(1.25,0) = 0.2865
however answer is 0.2774

the probability that two defective bricks are produced in any one hour?
poissonpdf(1.25,2) = 0.2238
however answer is 0.2305

the probability that at least one defective brick is produced in the next hour?
not sure about this one

P.S

The probability of k events in n trials has binomial distribution, even if for 'rare events' the Poisson distribution [computationallly less problematic...] gives accetable precision. The probability to have k defective briks in a stock of n, if p is the probability of a single defective brick, is...

$P(k,n)= \binom{n}{k}\ p^{k}\ (1-p)^{n-k}$ (1)

If p=.05 and n=25 the (1) supplies $P(0,25)= .277389573...$ and $P(2,25)=.23051765079...$ . The probability of least one defective brick in one hour is of course $1-P(0,25)= .722610426878...$

Kind regards

$\chi$ $\sigma$
August 31st 2011, 03:54 PM
Paymemoney

Re: Poisson Distribution question

but doesn't any events that occur at random with a constant average per some unit such as time and length or area is modeled as a Poisson distribution?
September 1st 2011, 02:03 AM
chisigma

Re: Poisson Distribution question

Quote:

Originally Posted by Paymemoney

but doesn't any events that occur at random with a constant average per some unit such as time and length or area is modeled as a Poisson distribution?

The real probability distribution of such type of process is binomial, even if in many pratical situations the Poisson distribution gives an excellent approximation. Binomial and Poisson distributions give pratically the same results when is $\lambda = p\ n <<1$ . In Your case is $\lambda= p\ n = 1.25$ , so that the results are slighly different. Here You can observe the values of...

$P_{b} (k,n) = \binom {n}{k}\ p^{k}\ (1-p)^{n-k}$ (1)

$P_{p}(k,\lambda)= \frac{\lambda^{k}\ e^{-\lambda}}{k!}$ (2)

... computed for k from 0 to 9 with $\lambda= p\ n= 1.25$ ...

$k=0\ ,\ P_{b}= .2773895\ ,\ P_{p}= .2865048$

$k=1\ ,\ P_{b}= .3649863\ ,\ P_{p}= .358131$

$k=2\ ,\ P_{b}= .2305176\ ,\ P_{p}= .2238318$

$k=3\ ,\ P_{b}= .09301589\ ,\ P_{p}= .09326328$

$k=4\ ,\ P_{b}= .02692565\ ,\ P_{p}= .02914477$

$k=5\ ,\ P_{b}= .0059519866\ ,\ P_{p}= .0072861937$

$k=6\ ,\ P_{b}= .001044208\ ,\ P_{p}= .0015177957$

$k=7\ ,\ P_{b}= .0001491726\ ,\ P_{p}= .00027106375$

$k=8\ ,\ P_{b}= 1.76651758\ 10^{-5}\ ,\ P_{p}= 4.23537119\ 10^{-5}$

$k=9\ ,\ P_{b}= 1.75618707\ 10^{-7}\ ,\ P_{p}= 5.88246\ 10^{-6}$

The most evident difference from the two is the fact that, for 'large' value of k, the $P_{b}$ decreases much steeper than the $P_{p}$ and the reason of that is obvious...

Kind regards

$\chi$ $\sigma$