Proving completeness of the average of a random normal sample Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show that $T(Y)=\frac{1}{n}\sum_iY_i$ is complete:
$$
h\text{ being a function s.t. }E_\theta(h(T))=0 \forall \theta\in\Theta\implies h\equiv 0.
$$
Because $T\sim N(\theta,\frac{1}{n})$, I have written out
$$
E_\theta(h(T))=\int_{-\infty}^{\infty}\sqrt{\frac{n}{2\pi}}\exp\left(-\frac{n}{2}(t-\theta)^2\right)h(t) \, dt.
$$
The book I'm using hints that I should look at the Laplace transform but I'm not sure how to proceed. Thank you for your help.
Edit: the wiki article on completeness has this example and also alludes to a solution using the Laplace transform. If someone can fill in the steps so I can learn how these things are done, I'll really appreciate it.

...or, consider your integral being the squared integral of (exp(-r^2)dr, from 0 to Infinity) in polar coordinates which can be solved easely. The result is sqrt{1/2*sqrt(pi)*erf (Infinity)}. erf(infinity) is a fancy way of saying "1".
