areal probability distribution (not sure if this is the right terminology)

Discussion in 'Scientific Statistics Math' started by Dan Lenski, Nov 17, 2010.

  1. Dan Lenski

    Dan Lenski Guest

    Hi all,
    I'm trying to understand how to derive the probability of a specific
    area being affected by a process which occurs with a known rate per
    unit area. Part of my problem may have to do with not knowing the
    correct terminology to describe this...

    For example, suppose that during a brief rain storm, my patio receives
    an average of 1 rain drops per cm^2. Now, how do I calculate the
    probability that a tile of area 10 cm^2 has *zero* rain drops land on
    it?

    rate: r = 1/cm^2
    area: A = 10 cm^2

    I believe the correct answer is: P=k * exp[ -r * A ]=k * exp[-10].

    I base this on the idea that no rain drop can land in any single cm^2
    of the tile, and that since the tile can be divided into 10
    independent regions of 1 cm^2, the probability of the entire tile
    remaining free of raindrops is X^10, where X is the probability that
    no rain lands on an individual subregion of area 1 cm^2.

    I'm not sure what the pre-factor k should be. I have a feeling this
    is a well-known problem in probability but can't remember what it's
    called. Any pointers in the right direction, or how to figure out the
    prefactor?

    Thanks!

    Dan
     
    Dan Lenski, Nov 17, 2010
    #1
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  2. Dan Lenski

    Ray Koopman Guest

    That sounds like you're trying to say that the number of drops
    in 1 cm^2 follows a Poisson distribution with mean r:

    P_1(n) = exp(-r) * r^n / n! .

    The sum of A independent Poisson variables with the same mean r is
    also a Poisson variable, with mean = A*r. So the probability of n
    drops in A cm^2 is

    P_A(n) = exp(-A*r) * (A*r)^n / n! ,

    and if r = 1 then P_10(0) = exp(-10).
     
    Ray Koopman, Nov 18, 2010
    #2
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  3. Dan Lenski

    Rod Guest

    There is a difference if you pre-select a tile or just say any tile on my
    patio.
     
    Rod, Nov 18, 2010
    #3
  4. Dan Lenski

    Dan Lenski Guest

    Ah yes, thanks! The Poisson distribution is exactly what I'm looking
    for, just couldn't figure out the name. So the prefactor turns out to
    be... one. Excellent.

    Dan
     
    Dan Lenski, Nov 19, 2010
    #4
  5. Dan Lenski

    Dan Lenski Guest

    Right, sorry if I wasn't clear on this. I meant the case of picking
    one specific tile and finding no drops on it, not of finding one tile
    among many with no drops on it.

    Ray's answer of the Poisson distribution was just what I was looking
    for.

    Dan
     
    Dan Lenski, Nov 19, 2010
    #5
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