The problem involves analyzing a communication system where four messages, $X_1, X_2, X_3, X_4$, are transmitted over a channel with interference, resulting in four received signals, $Y_1, Y_2, Y_3, Y_4$. The table shows the probabilities $P(Y_i | X_j)$ for each pair of transmitted and received messages. Here’s the approach to solve the problem:

Steps to Solve

1. Interpret the table values: The values in the table represent conditional probabilities $P(Y_i | X_j)$, the probability of receiving $Y_i$ given that $X_j$ was sent.

2. Define the Problem: We need to determine specific probabilities, such as the probability of correctly receiving each message, or the overall probability of an error in transmission. This typically involves calculating marginal and joint probabilities.

3. Calculate Marginal Probabilities: If the probabilities $P(X_j)$ for each $X_j$ are not provided, you may need to assume they are equally likely. This means $P(X_j) = \frac{1}{4}$ for each $j$ if there is no other information given.

4. Find Error Probabilities:

   • The probability of error for a specific $X_j$ could be calculated as $1 - P(Y_i = X_j | X_j)$.
   • For example, for $X_1$, the correct reception is $Y_1$. So, $P(\text{error for } X_1) = 1 - P(Y_1 | X_1)$.

5. Total Probability of Error: Sum the individual probabilities of receiving the wrong message, possibly using the law of total probability if needed.

Would you like to proceed with a specific calculation, such as finding the total error probability or any specific conditional probability?

Relative Questions

1. What is the probability of receiving $Y_1$ given that $X_2$ was transmitted?
2. How would you calculate the probability of a correct transmission for each $X_j$?
3. What is the importance of knowing $P(Y_i | X_j)$ in a communication system with interference?
4. How can we determine the likelihood of errors in this communication setup?
5. Can this method be extended to more complex communication systems with additional messages and interferences?

Tip

When dealing with probabilities in a communication system, visualizing or mapping out the relationships between transmitted and received signals can simplify the calculation process and provide insights into potential sources of error.