Leave $ X_1, ldots, X_n $ ~ Poi ($ theta $). Calculate the moments estimators for. $ theta $ in the following cases:

Remember it $ E (X_1) = Var (X_1) = theta $

(a) $ hat { theta} _1 $ Equating the first non-central theoretical moment with the first non-central empirical moment.

(second) $ hat { theta} _2 $ equaling the theoretical variance with the empirical variance.

(do) $ hat { theta} _3 $ Equating the second non-central theoretical moment with the second non-central empirical moment.

I just want to know if I did it correctly. It seems too easy.

(a) $ hat { theta} _1 = overline {x} = frac {1} {n} sum ^ {n} _ {i = 1} x_i $

(second) $ hat { theta} _2 = line {x} = frac {1} {n} sum ^ {n} _ {i = 1} (x_i- mu_1) ^ 2 $

(do) $ hat { theta} _3 ^ 2- hat { theta} _3 = frac {1} {n} sum ^ {n} _ {i = 1} x_i ^ 2 $