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1. ## 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. taylor expansion on e^-lambda

3. Originally Posted by Juret
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.

SBR Founder Join Date: 8/28/2005

4. Originally Posted by Ganchrow
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.

SBR Founder Join Date: 8/28/2005

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

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