1. #1
    Juret
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    Solve for lambda (Poisson)

    Does anyone know how to solve for lambda in the Poisson equation below?

    P(X=k)=

    As of now, I'm using Excel's solver function to find the mean (lambda) given the market's odds for a Poisson distributed total.

    Thanks

  2. #2
    rsigley
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    taylor expansion on e^-lambda

  3. #3
    Ganchrow
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    Quote Originally Posted by Juret View Post
    Does anyone know how to solve for lambda in the Poisson equation below?

    P(X=k)=
    There's no closed-form solution.

    Analytically, what you're looking for is the Lambert W-function.

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  4. #4
    Ganchrow
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    Quote Originally Posted by Ganchrow View Post
    There's no closed-form solution.

    Analytically, what you're looking for is the Lambert W-function.
    So if you do the algebra:

    P*k! = λk * e
    (P*k!)1/k = λ*e-λ/k
    -(1/k)*(P*k!)1/k = -λ/k*e-λ/k


    Which would then allow you to use the Lambert W function directly. Specifically:

    λ = -k * W[-(1/k)*(P*k!)1/k]


    So provided you can find a suitable implementation of the Lambert W, then given any pair of input parameters, k and P,you can come up with arbitrarily precise solutions for λ.

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  5. #5
    Juret
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    Thanks guys, I'll take a look at this!

  6. #6
    aosipov
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    Can you give me specific example on how to solve this? W[-(1/k)*(P*k!)1/k]


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