Problem 2. Find P[K < E[K]] when K is Geometric (2/5). K is Binomial (8, 1/4). K is Poisson (6). K is discrete uniform (1, 11). a. b. c. d.

i-0¹¹ for a - 1, 2; but ₁0¹i 3? • In the case d = 1,2 what is the expected length of a path returning to the origin? . In the case d = 1,2, what is the probability of two returns? Three returns? An infinity of returns to the origin? 7

There are two rooms. Room 1 has 10 boxes, one of them has $10 in it and the other 9 are empty. Room 2 has 20 boxes in it, one of them has $20 in it and the other 19 are empty. Alice goes to room 1 and picks a box at random. Bob goes into room 2 and picks a box at random. Alice get X dollars, Bob gets Y dollars. Then which of the following is true? 1. E[X] # E[Y] and Var[X] = Var[Y]. 2. E[X] = E[Y] and Var[X] = Var[Y]. 3. E[X] E[Y] and Var[X] Var[Y]. 6. E[X] = E[Y] and Var[X] > Var[Y].

Problem 7: (4.10) Show that for any integer n ≥ 1, we have n(n + 1)(n + 2) 3 n Σk(k+ 1) = k=1

A flow of claims arriving at an insurance company is represented by a homogeneous Poisson process Nt in continuous time. (For now, we just count the number of claims arrived by time t.) Suppose that the mean inter-arrival time is equal to 1/λ, where A is a positive parameter. Let the unit of time be an hour. Question 8 What is E{N₁} ? (In N₁ the sub index 1 is a time (one hour).) 1 1² 1 1 Question 9 Are the r.v.'s N₁ and N₂ independent? (The sub index 2 in N₂ is a time (two hours).) Yes No Question 10 Are the r.v.'s N₁ and N₂ – N₁ dependent? Yes No

5.2.9

Question 10. Prove that if d₁, d2, d3,..., dr divide n (including 1 and n), then ø(d₁) + (dr) = n. o(d₂) + (d3) + ... +

5. Let A = {r € Z : r mod 7 = 2} and let B = {r EZ: 7|(x – 2)}. Show that A = B. %3D