We will show in the next chapter that if x1 and x2 are independent with xi ∼ n (µi, σ2 i ), then x1 + x2 ∼ n (µ1 + µ2, σ2 1 + σ 2 2). use this result to find p(x < y ) for x ∼ n (a, b), y ∼ n (c, d) with x and y independent. hint: write p(x < y ) = p(x − y < 0) and then standardize x − y . check that your answer makes sense in the special case where x and y are i. i.d.