GATE Electronics and Communications (EC) 2018 Solved Paper

A random variable 𝑋 takes values −0.5 and 0.5 with probabilities $\frac{1}{4}$ and $\frac{3}{4}$, respectively. The noisy observation of 𝑋 is 𝑌 = 𝑋 + 𝑍, where 𝑍 has a uniform probability density over the interval (−1, 1). 𝑋 and 𝑍 are independent. If the MAP rule-based detector outputs $\hat{\mathit{X}}$ as

then the value of 𝛼 (accurate to two decimal places) is _______.