To determine the minimum decision error probability for a communication system, we need to apply the Maximum A Posteriori (MAP) decision rule. This rule helps the receiver make the most optimal decision about the transmitted bit (0 or 1) based on the received signal, thereby minimizing the overall probability of error. The decision is made by comparing the posterior probabilities of each bit given the received value, or equivalently, by comparing the product of the prior probability and the conditional probability density function (PDF).

Prior Probabilities of Emitted Bits

First, we identify the prior probabilities of the source emitting each bit. These are given in the problem statement:

• Probability of the source emitting bit 0: \(P(0) = \frac{1}{3}\)
• Probability of the source emitting bit 1: \(P(1) = \frac{2}{3}\)

Conditional Probability Density Functions

The problem provides the conditional probability density functions (PDFs) of the received value \(R\), given which bit was transmitted. These functions describe the distribution of the received signal for each possible transmitted bit:

• If bit 0 was sent, the conditional PDF of \(R\) is:

  \(f_{\left(R/0\right)}\left(x\right) = \begin{cases} \frac{1}{4}, & -3 \le x \le 1 \\ 0, & \text{otherwise} \end{cases}\)

  This is a uniform distribution over the interval \([-3, 1]\).

• If bit 1 was sent, the conditional PDF of \(R\) is:

  \(f_{\left(R/1\right)}\left(x\right) = \begin{cases} \frac{1}{6}, & -1 \le x \le 5 \\ 0, & \text{otherwise} \end{cases}\)

  This is a uniform distribution over the interval \([-1, 5]\).

Decision Rule and Regions for Minimum Error

According to the MAP decision rule, the receiver should choose the bit \(i\) that maximizes the product \(P(i) \cdot f_{(R/i)}(x)\) for a given received value \(x\). Let's calculate these products for both bits:

• For bit 0:

  \(P(0) \cdot f_{(R/0)}(x) = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}\) for \(-3 \le x \le 1\), and \(0\) otherwise.

• For bit 1:

  \(P(1) \cdot f_{(R/1)}(x) = \frac{2}{3} \times \frac{1}{6} = \frac{2}{18} = \frac{1}{9}\) for \(-1 \le x \le 5\), and \(0\) otherwise.

Now, we define the decision regions by comparing these weighted likelihoods across the possible range of \(x\):

• Case 1: \(x \in [-3, -1)\)

  In this range, only \(P(0) \cdot f_{(R/0)}(x) = \frac{1}{12}\) is non-zero, because \(f_{(R/1)}(x)\) is \(0\) for \(x < -1\). Therefore, for any \(x\) in this interval, the receiver decides 0.

• Case 2: \(x \in [-1, 1]\)

  In this overlapping range, both weighted likelihoods are non-zero. For bit 0, the value is \(\frac{1}{12}\), and for bit 1, the value is \(\frac{1}{9}\). Since \(\frac{1}{9} > \frac{1}{12}\), for any \(x\) in this interval, the receiver decides 1.

• Case 3: \(x \in (1, 5]\)

  In this range, only \(P(1) \cdot f_{(R/1)}(x) = \frac{1}{9}\) is non-zero, because \(f_{(R/0)}(x)\) is \(0\) for \(x > 1\). Therefore, for any \(x\) in this interval, the receiver decides 1.

Based on this analysis, the optimal decision regions are:

• Region to decide 0 (\(R_0\)): \([-3, -1)\)
• Region to decide 1 (\(R_1\)): \([-1, 5]\)

Calculating Minimum Error Probability

The total minimum decision error probability, \(P_e\), is the sum of probabilities of making incorrect decisions:

\(P_e = P(\text{decide } 0 | \text{sent } 1) P(\text{sent } 1) + P(\text{decide } 1 | \text{sent } 0) P(\text{sent } 0)\)

Let's calculate each term:

• Probability of Deciding 0 when bit 1 was sent:

  This error occurs if the received value \(x\) falls into region \(R_0\) (decide 0) but bit 1 was actually transmitted. We calculate this as \(\int_{R_0} f_{(R/1)}(x) dx\).

  For \(R_0 = [-3, -1)\), the conditional PDF \(f_{(R/1)}(x)\) is \(0\). Therefore, \(P(\text{decide } 0 | \text{sent } 1) = \int_{-3}^{-1} 0 \, dx = 0\).

  The contribution of this error type to the total error probability is: \(0 \times P(1) = 0 \times \frac{2}{3} = 0\).

• Probability of Deciding 1 when bit 0 was sent:

  This error occurs if the received value \(x\) falls into region \(R_1\) (decide 1) but bit 0 was actually transmitted. We calculate this as \(\int_{R_1} f_{(R/0)}(x) dx\).

  For \(R_1 = [-1, 5]\), the conditional PDF \(f_{(R/0)}(x)\) is non-zero only for \(x \in [-1, 1]\), where it is \(\frac{1}{4}\). Outside this range (i.e., for \(x \in (1, 5]\)), \(f_{(R/0)}(x)\) is \(0\).

  So, \(P(\text{decide } 1 | \text{sent } 0) = \int_{-1}^{1} \frac{1}{4} dx = \frac{1}{4} [x]_{-1}^{1} = \frac{1}{4} (1 - (-1)) = \frac{1}{4} (2) = \frac{1}{2}\).

  The contribution of this error type to the total error probability is: \(P(\text{decide } 1 | \text{sent } 0) \times P(0) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).

The minimum decision error probability \(P_e\) is the sum of these contributions:

\(P_e = 0 + \frac{1}{6} = \frac{1}{6}\)

Thus, the minimum decision error probability for this communication system is \(\frac{1}{6}\).