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A source emits bit $0$ with probability $\frac{1}{3}$ and bit $1$ with probability $\frac{2}{3}$. The emitted bits are communicated to the receiver. The receiver decides for either $0$ or $1$ based on the received value $R$. It is given that the conditional density functions of $R$ are as $$f_{R \mid 0}( r) = \begin{cases} \frac{1}{4}, & -3 \leq x \leq 1, \\ 0, & \text{otherwise,} \end{cases} \text{ and } f_{R \mid 1}( r) = \begin{cases} \frac{1}{6}, & -1 \leq x \leq 5, \\ 0, & \text{otherwise,} \end{cases}$$ 

The minimum decision error probability is

  1. $0$
  2. $1/12$
  3. $1/9$
  4. $1/6$
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