Decision rule that minmize the probability of error

Question

Given

We consider a real-valued, discrete-time communication system with a channel gain $h$ and additive white Laplacian noise of unit scale with two possible signals $s \in (-\mu,+\mu)$ that are sent equiprobably: $y = hs + w$. To be more precise, the noise $W$ has the pdf

\begin{align} f_W(w) = \frac{1}{2}e^{-|w|} \end{align}

Further, we define $SNR = \mu$, as a signal-to-noise ratio for noise of unit scale.

Question

Suppose first $h = 1$. Specify the decision rule that minimizes the probability of error in deciding on the transmitted message $s$, and compute the average error probability $P_e$ in terms of our definition of SNR.

Attempted solution

For $h = 1$ we have that

\begin{align} y = s + w \end{align}

We will have two cases, first for $-\mu$

\begin{align} y = -\mu + w \\ P(y|-\mu) = \frac{1}{2}e^{-|w + \mu|} \end{align} and call this hypothesis $H_0$

Secondly, for $+\mu$ we will have

\begin{align} y = \mu + w \\ P(y|+\mu) = \frac{1}{2}e^{-|w-\mu|} \end{align} and this hypothesis is called $H_1$

My question is, do I always have to calculate conditional probability when when calculating the probability of error? And I also wonder that is the equation for the probability of error in this case?

Answer 1

First question: Yes, When dealing with error probabilities you always have to calculate the conditional probabilities.