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One sided testing of two Poisson distributions?

  1. Feb 11, 2008 #1
    I want to test if one Poisson distributed result a is large than another one b.
    I don't know much about statistics, but I understood the Wiki article about testing normal distribution however they need the number of samples there.

    Basically I measure two Poisson distributed variables, I get two values and want to know the probability that one is larger than the other.

    Can someone give my a quick reference (online or good book), where I can find my problem as close as possible?
    Last edited: Feb 11, 2008
  2. jcsd
  3. Feb 11, 2008 #2


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    You can compute the difference between the two Poisson means and see whether the difference is significantly different from zero under the Skellam distribution.

    Or (with a large enough sample) you can assume that the normal distribution will be a reasonable approximation.
    Last edited: Feb 11, 2008
  4. Feb 11, 2008 #3
    I think I just about know what to do with the Skellam. But it has a funny Bessel function.
    I have both means approx. 500.
    What would be the method with the normal distribution?
  5. Feb 11, 2008 #4


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    If you know the true variances, the z test.

    If you are using computed variances, then technically you should use t test with equal or unequal variances, as the case may be. As the sample size increases, the z-test becomes a good approximation to the t-test (e.g. for n > 40).
  6. Feb 11, 2008 #5
    I tried to look through these tests, but I'm not sure what to take. I only know that I measured a single value
    x and single value y. Both are supposed to be Poisson (so I expect x+- sqrt(x) and y+-sqrt(y)).

    In this case I'm not sure how interpret "computed variance" or "sample size".
    I know about mathematics, but not of the formalities of statistics :(
  7. Feb 12, 2008 #6
    I found the following (attachment) in
    "An improved approximate two-sample poisson test" (M.D.Huffman)

    Just to make sure I got it right and plug in the right values:
    I use [itex]\alpha=0.05, p=0.90[/itex] as sensible values?
    I look up z in a table? (i.e. [itex]z_{0.95}=1.65, z_{0.90}=1.28[/itex])
    I estimate [itex]\varrho[/itex] from initial measurements.
    Should I use equal counting time [itex]d=1[/itex] for best results?
    By equation (4) I will find how long I have to measure...

    Attached Files:

  8. Feb 12, 2008 #7


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    In a z-test you are assumed to know both means and variances. Poisson is a one-parameter distribution (say k) where mean is a function of k, and variance is also a function of k; so you can derive both means and both variances if you know the k parameter for each of the distributions. Put differently, if you know the mean then you know the variance.
    Last edited: Feb 12, 2008
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