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

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

 

[safina]

Notice that the quantity you want to estimate is

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

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 $$\kappa\left(\theta\right)$$.