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

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...

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

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?

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 $\displaystyle \lambda = p\ n <<1$. In Your case is $\displaystyle \lambda= p\ n = 1.25$, so that the results are slighly different. Here You can observe the values of...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$