Statistics: Point estimation

Question

If we have a sample of $x=2$ from a $Po(6 \cdot \lambda)$ distribution.

How do we calculate $\lambda*$ and $d(\lambda*)$?

I think that $\lambda* = \frac{2}{6} = \frac{1}{3} $ but I am not sure about the other one.

Answer 1

Whatever you may mean by $\lambda *$ and $d(\lambda *),$ which I suppose are defined in your text or notes, the following information may be helpful.

If $X_1$ and $X_2$ are independently $\mathsf{Pois}(6\lambda),$ Then $$S = X_1 + X_2 \sim \mathsf{Pois}(12\lambda).$$ Then $E(S) = 12\lambda$ and $Var(S) = 12\lambda.$

So $S$ is a point estimate of $12 \lambda$ and $S/12$ is a point estimate of $\lambda.$