Solve for lambda (Poisson)

[Juret]

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

[rsigley]

taylor expansion on e^-lambda

[Ganchrow]

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.

[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! = λ^k * e^-λ
(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.

[Juret]

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

[aosipov]

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