Need help answering this practice problem.

4. Consider a digital communication system, where two bits of binary messages are transmitted at a time. This is done by mapping the two bits to one of the four signal points in the 2- dimensional plane. Specically, when S1 = (0,0) is transmitted, the transmitted signal point is (X1,X2) = (1,1).When $2 = (0,1) is transmitted, (X1,X2) = (1,1). When 53 = (1,1), (X1,X2) = (1,1). When 54 = (1,0), (X1,X2) = ( 1,1). (Refer to the diagram below.) It is assumed that the received signal (Y1, Y2) are corrupted by normal noise as Y1 = X1 + N1 Y2 = X2 + N2 where N1 and N2 have mean 0 and variance 0'2. They are independent of each other. (a) Given that $1 is transmitted, write the marginal PDF of Y1 and the X2 joint PDF of (Y1, Y2). (b) The receiver decides that 51 was transmitted when Y1 2 0 and 54 O 051 Y2 2 0. Given that 51 was transmitted, calculate the probability of an erroneous decision. (c) In general, for two events A and B, show that P(A U B) S P(A) + P(B). (This is called the union bound.) (d) Assuming that 51 was transmitted, note that an error occurs if Y1