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A random variable $X$ takes values $-0.5$ and $0.5$ with probabilities $\dfrac{1}{4}$ and $\dfrac{3}{4}$, respectively. The noisy observation of $X\:\text{is}\:Y=X+Z,$ where $Z$ has uniform probability density over the interval $(-1,1).\: X$ and $Z$ are independent. If the MAP rule based detector outputs $\hat{X}$ as
$$\hat{X}=\left\{\begin{matrix} -0.5, & Y<\alpha \\ 0.5,& Y\geq \alpha, \end{matrix}\right.$$ then the values of $\alpha$ (accurate to two decimal places) is ________.
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