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A random variable $\text{X}$ takes values $-0.5$ and $0.5$ with probabilities $\frac{1}{4}$ and $\frac{3}{4}$, respectively.

The noisy observation of $\text{X}$ is $Y=X+Z,$ where $\text{Z}$ has uniform probability density over

the interval $\text{(-1.1)}$. $\text{X}$ and $\text{Z}$ are independent. If the $\text{MAP}$ rule based detector outputs $\widehat{X}$ as

$$\widehat{X}=\left\{\begin{matrix} -0.5, & Y<\alpha \\ 0.5,& Y\geq \alpha, \end{matrix}\right.$$

then the values of $\alpha$ (accurate to two decimal places) is ________.

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