MLE of Poisson Dist: Find \lambda^2+1

[mrkb80]

Homework Statement

Let X_1,...,X_n be a random sample from a poisson distribution with mean \lambda

Find the MLE of \lambda^2 + 1

Homework Equations

The Attempt at a Solution

I found \hat{\lambda}=\bar{x}

Can I just square it and add 1 and solve for lambda hat?

If not I have no idea how I would get the FOC (with respect to \lambda^2 + 1)

of the log-likelihood function \ln{L(\lambda^2+1)}=-n\lambda + \Sigma_{i=1}^n x_i \ln{\lambda} - \ln{\Pi_{i=1}^n x_i!}

 

[haruspex]

I found \hat{\lambda}=\bar{x}

Can I just square it and add 1 and solve for lambda hat?

That's my understanding of how MLE works. If α is the value of λ that maximises the likelihood of the observed data, then (α²+1) must be the value of λ²+1 that does the same.

 

[mrkb80]

cool. thanks again.