please share complete step by step solution. I need complete solution

= In digital communication systems, the main objective is to minimize the probability of bit error when a binary symbol S is transmitted over a noisy communication channel, where S; = 0 and S2 = 1. The most simple noisy channel model is called Additive White Gaussian Noise (AWGN) model shown in the figure below Y N where N is the channel noise which is modeled as unit-variance Gaussian random variable with zero mean, and Y is the channel output given by Y = S + N Eq. (1) Furthermore, it is assumed that the transmitter emits binary symbol "0" with probability P[S, = 0] =p Eq. (2) Given the above system model, the objective is to design an optimum binary symbol detector. Thus, the signal detection problem can be stated as follows: "Given the observation Y, perform a mapping from Y to an estimate $ of the transmitted symbol, Si, in a way that would minimize the probability of error in the decision-making process", where the error probability is defined as Pe [S1Y] = P[S, is not sent|Y] = 1-P[Ssent|Y] Eq. (3) Tasks 1. Determine the probability, P[S2 = 1]. 2. Determine the Probability Density Function (PDF) of the random variable Y. 3. Mathematically formulate the statement of the signal detection problem and determine the optimum decision threshold. 4. Evaluate the error probability, P. [S, IY)