Question

iestion There are n students enrolled in Math 1A in Semester 1, each passes with probability p independently of each of other students. Each student who passes Math 1A will enrol into in Seme that subject with probability p, once again independently of other students. Let X be the total number of students passing Math 1A, and Y be the total number of students passing Math 1B. (a) Determine the probability mass function fy for Y. (b) Are X and Y independent? As always, justify your answer

Answer 1

X sim Binomial (m, p) passing(see, each student is a Random variable so, passing of students amoung all will be Binomial) Math 1A Math 1 x) = (x y) p^y(1 - p) ^x - y, y = 0(1) x so, f(x, y) = f(x) f(y x) = (n x) p^x(1 - p) ^n - x (x y) p^y(1 - p) ^x - y = (n x) (x y) p^x + y (1 - p) ^n - y, y = 0(1) x x = 0(1) n (a) joint pdf (b) see that, when x in given, y must be can that. X. so, y in always dependent on x. Hence, they are not independent