# rearranging gamma functions to solve Weibull distribution parameters

I'm trying to generate a Weibull distribution to see if it fits some data. I have an arithmetic mean and variance for this data but I'm struggling to analytically create the specific Weibull parameters from these.

For a Weibull distribution:

μ=λΓ(1+(1/k))

σ^2=λ^2 [Γ(1+2/k)-(Γ(1+1/k))^2 ]

I have found that these equations rearrange to:

(σ^2/μ^2)+1=(2/k)!/((1/k)!)^2

This can be solved by Wolfram Alpha and with values of μ=7.3151 and σ=4.0730 this gives k=1.8648 and λ=8.2383.

The problem that I'm facing is that I don't understand the jump made by Wolfram from the rearranged equation to the answer. If anyone could offer an explanation that would be great.

• You cannot obtain a unique solution for two variables from the single last equation. Generically, though, two equations in two unknowns have a discrete number of solutions, typically just one. Could you therefore expand on what you are looking for in way of an "explanation"?
– whuber
Apr 11 '17 at 16:31
• Wolfram gives me 3 solutions to this equation. Two negative and one positive. The positive is the only useful solution. By "explanation" I mean that I would like to know how those answers are obtained. Does Wolfram calculate these iteratively or is there an analytical way to solve these? Apr 11 '17 at 18:09