Determine the average probability of error as a function

Question

Response to the following problem:

In a binary PAM system, the input to the detector is

y_m = a_m + n_m + i_m

where a_m = ±1 is the desired signal, nm is a zero-mean Gaussian random variable with variance σ²n , and im represents the ISI due to channel distortion. The ISI term is a random variable that takes the values -1 2 , 0, and ½ with probabilities ¼ , ½ , and ¼ , respectively. Determine the average probability of error as a function of σ²n .