# Showing family is NOT complete

How would I show that $$f(x;\theta) = \frac1{2\theta}$$ where $x$ is between positive and negative $\theta$ and $\theta$ is between $0$ and $\infty$ is NOT a complete family?

I know that I need to find a non-zero function $u(x)$ whose expectation will be $0$, but I am struggling with finding this function.

Thanks

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Let $X$ be a random variable whose distribution is known to be one of these. Then by symmetry $E(X)=0$. –  André Nicolas Feb 6 '12 at 8:46

## 1 Answer

Hint: try a function $u(x)$ which is odd, i.e., with $$u(-x) = -u(x).$$

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If I select $U(x)=x^3$this works right? –  clarkson Apr 22 at 3:19
Then the integration=0 but it does not require u(x) to be zero right? –  clarkson Apr 22 at 3:41
@clarkson: all the expansion coefficients are 0 but the function is not zero so the family is not complete. –  Fabian Apr 23 at 12:31