GATE Electronics and Communications (EC) 2019 Solved Paper

A random variable $\mathit{X}$ takes $-1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $\mathit{N}$, so that the random variable at the channel output is $\mathit{Y}=\mathit{X}+\mathit{N}.$ The noise $\mathit{n}$ is independent of $\mathit{X}$, and is uniformly distributed over the interval $\left[-2,2\right].$ The receiver makes a decision Where the threshold $\mathit{\theta }\in \left[-1,1\right]$ is chosen so as to minimize the probability of error $\mathit{P}\mathit{r}\left[\stackrel{\wedge }{\mathit{X}}\ne \mathit{X}\right].$ The minimum probability of error, rounded off to 1 decimal place, is ______________.