Show that statistic is (not) sufficient

Question

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$

Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$

$$f_{X|T}(x|t) = \frac{exp(-1/2(x-\mu)^2)}{exp(-1/2(x-\mu)^2)+exp(-1/2(x+\mu)^2)}I(t=|x|) = \frac{exp((x+t)\mu)}{1+exp(2t\mu)}I(t=|x|)$$

Bt how can I prpve that it does deppends on $ \mu$ or not? For me, it does, but I am having troubles to write a good argument. Should I, for example, change t for |x|?

Thanks in advance!

Answer 1

For example, $f_{X\mid T}(t\mid t)=1/(1+\mathrm e^{-2t\mu})$ if $t\ne0$ hence $f_{X\mid T}(\ \mid t)$ does depend on $\mu$ except if $t=0$.