GATE EC 2022 Set 1 Q53 — Digital Communications

Consider a channel over which either symbol $x_{A}$ or symbol $x_{B}$ is transmitted. Let the output of the channel $Y$ be the input to a maximum likelihood $\text{(ML)}$ detector at the receiver. The conditional probability density functions for $Y$ given $x_{A}$ and $x_{B}$ are:

$$f_{Y|x_{A}}(y) = e^{-(y+1)}u(y+1),$$

$$f_{Y|x_{B}}(y) = e^{(y-1)} (1-u(y-1)),$$

where $u(\cdot)$ is the standard unit step function. The probability of symbol error for this system is _____________ (rounded off to two decimal places).