1. Problem 1

2. Problem 2

4. Problem 3

7. Problem 4

9. Problem 5

10. Problem 6

12. Problem 7

13. The observation vector x has two elements x1 and x2 that are defined by
Define new Gaussian random variable Y whose sample vlaue y is equal to the difference between x1 and x2,
Thus we have

14. Suppose we know symbol 0 was transmitted, so
Since the condition x1>x2, y>0, corresponds to the receiver making a decision in favor of 1, we deduce that the conditional probability of error, so

15. So p10 is equal to
The average probability of bit error is (assuming equiprobable symbols)

16. Problem 8