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I would like to derive the following and need some help. Thanks in advance.

Suppose a continuous variable $x$ has the following distribution, $x\sim N^{+}(0,\sigma_L^2)\; if\; x>0$ and $x\sim N^-(0,\sigma_H^2)\; if\; x<0$, where $\sigma_L^2<\sigma_H^2$. In other words, the conditional variance of $x$ when it is negative is larger than when it is positive.

Now we receive a signal on $x$ and it is $s=x+\epsilon$, where $\epsilon \sim\mathcal{N}(0,\sigma_\epsilon^2)$.

I would like to derive the conditional probability function of $x$ conditional on the signal $s$, i.e. $f(x|s)$, and also the conditional expectation $E[x|s]$ and conditional variance $Var[x|s]$. Could you help with this?

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    You conditional distribution seems ill-defined. The distribution of $x$ seems to be conditioned on the value of $x$ itself... – 2017-01-30
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    I have edited. I think it should be well-defined. The pdf of $x$ is the combination of two half normals with different variances. Thanks. – 2017-01-30

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