Please complete both problems fully please

Problem 1 (5 points). The stock price of a company is a dynamic variable that can change from day to day. In this particular case, we know that if the stock price is $X on a certain day, then the next. day it can either increase by $1 (to become $X+1) or decrease by $1 (to become $X1), each with a probability of %. Suppose that today the stock price of this cmnpany is $100, and you purchased a full share of it. Based on this information, calculate the probabilities of the following events: (a) On day 30 (after the purchase), you will have a prot of $10. (b) On day 30, you will lose $ 5. (c) On day 30, the stock price remains $100. Hint: think of Binomial distribution. Problem 2 (5 points). Let X1, ,Xn be independent Exp(/\\) random variables. For each 1 3 m. g n, dene Sm : 2:, X. Note that Sm m Ganunafm A) for each m _1 ..... _n. Moreover, for each 1 g m. g n, IP(Sm S t t) P(Sm+1 > t) for all t > 0. Using this, prove that _,,(At)m for all t > 0. m! IP09?\" S t 0 0 otherwise. Then use integration by parts