1. ## Poisson random variable?

For determining the half=lives of radioactive isotopes, it is important to know what the background radiation is in a given detector over a period of time. Data taken in a y-ray detection experiment over 300 one-second intervals yielded the following data:

Data......Frequency
0 ............. 17
1 ............. 47
2 ............. 63
3 ............. 63
4 ............. 49
5 ............. 28
6 ............. 21
7 ............. 11
8 .............. 1

a) Do these look like observations of a Poisson variable with mean $\lambda=3$?

I calculated the sample mean by: (0*17+1*47+2*63+3*63+4*49+5*28+6*21+7*11+8*1)/300=303/100=3.03

and sample variance by:
$
(0-3.03)^{2}*(17/300)+(1-3.03)^{2}*(47/300)+...+
$
(8-3.03)^{2}*(1/300)=43744099/12000000=3.18

b) Construct a probability histogram with $\lambda=3$ and a relative frequency histogram on the same graph. I'm not sure how to find the values for the histogram. I am not sure how to read the poisson table because the amounts I calculate are different than the ones on the chart when I use $(\lambda^{x}e^{-\lambda})/x!$

2. For the probability with $\lambda=3$, for 0....8 I got .049, .149, .224, .224, .168, .1, .05, .02, .008 by using the calculation $\lambda^{x}e^{-\lambda}/x!$.

Is this the correct way to calulate the probability? I tried using the number off the Poisson table, but it didn't seem to turn out right.

3. Originally Posted by antman
For the probability with $\lambda=3$, for 0....8 I got .049, .149, .224, .224, .168, .1, .05, .02, .008 by using the calculation $\lambda^{x}e^{-\lambda}/x!$.

Is this the correct way to calulate the probability? I tried using the number off the Poisson table, but it didn't seem to turn out right.
Are you expected to do a goodness-of-fit test of your data against the Poisson distribution?

You have used the correct approach to calculate the above probabilities, assuming a Poisson distribution. I suppose that if you're not expected to do the above test, then you'll just have to compare those probabilities against the actual probabalities and use your own best judgement:

The total frequency count is 300. So Pr(count = 0) = 17/300 = 0.057, Pr(count = 1) = 47/300 = 0.157 etc.

4. I am supposed to answer if these look like observations of a Poisson random variable with mean and answer the other questions in order to come to a conclusion.

Since my relative frequency graph is very similar to the probablility graph of $\lambda=3$ and the mean and variance are so close, I think the data do look like observations of a Poisson random variable with mean . I hope that is enough characteristics. I just based it in what I have read so far.

As for the graph, I am not sure how to create a histogram so I can enter data for 0. The first data that can be represented by the x-axis on the excel graphs is for 1. Rather than 0 to 8, it looks as if I am entering the probability values for 1 to 9.