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A random variable X takes values -1 and +1 with probabilities 0.2 and 0.8, respectively. It is transmitted across a channel which adds noise N, so that the random variable at the channel output is Y=X+N. The noise N is independent of X, and is uniformly distributed over the interval [-2, 2]. The receiver makes a decision ̂X=-1, if Y≤θ and ̂X=+1, if Y>θ where the threshold θ∈[-1,1] is chosen so as to minimize the probability of error Pr[̂X≠X]. The minimum probability of error, rounded off to 1 decimal place, is
Correct : 0.1
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