Unbiased estimator of a function

[safina]

Homework Statement

For a random sample X_{1}, ..., X_{n} from the Poisson distribution, find an unbiased estimator of \kappa\left(\theta\right) = \left(1 + \theta) e^{-\theta}.

The Attempt at a Solution

I know that the pmf of Poisson distribution is f\left(x; \theta\right) = \frac{e^{-\theta}\theta^{x}}{x!} I_{(o, 1, ...)}\left(x\right)
The parameter is \theta, but I do not know how to find the unbiased estimate of this problem.

 

[statdad]

Notice that the quantity you want to estimate is

<br /> P(X \le 1) = P(X=0) + P(X = 1) = e^{-\theta} \frac{\theta^0}{0!} + e^{-\theta} \frac{\theta^1}{1!} = e^{-\theta} + e^{-\theta} \theta = e^{-\theta}\left(1 + \theta\right)<br />

 

[safina]

Notice that the quantity you want to estimate is

<br /> P(X \le 1) = P(X=0) + P(X = 1) = e^{-\theta} \frac{\theta^0}{0!} + e^{-\theta} \frac{\theta^1}{1!} = e^{-\theta} + e^{-\theta} \theta = e^{-\theta}\left(1 + \theta\right)<br />

Okey. Thank you statdad for your reply.
But, I will be very happy if you will tell me how to find an unbiased estimator of this \kappa\left(\theta\right).