i know how to do the first prat for question 7. But i do not know how to deal with the second part. Also, the proof of question 8 is hard for me as well. 

7. Prove the "memoryless property" of Exponential distribution and geometric distribution: let X ~ Exponential(), Y ~ Geometric(p). Then, P(X > s + t| X > s) = P(X > t), Vs, t> 0; (1) P(Y-12k+m|Y -12 m) = P(Y > k), V non-negative intergers k, m. (2) 8. If a random variable Y only takes values in {1, 2, 3, ...} and satisfies (2), prove Y must satisfies geometric distribution for some 0 0. Hint: First try to find out the value of P(X > x) for all integers; then try to find out the value of P(X 2 x) for all rational numbers x; finally, use the fact P(X > x) is a non-increasing function