Sheila is testing model drones. She starts with n model drones which each independently have probability p of flying successfully each time they are flown, where 0 < p < 1. Each day, she flies every single drone and keeps the ones that fly successfully (i.e. don't crash), throwing away all other models. She repeats this process for many days, where each "day" consists of Sheila flying any remaining model drone and throwing away any that crash. Let Xi be the random variable representing how many model drones remain after i days. Note that X0 = n. Justify your answers for each part.

(a) What is the distribution of X1? That is, what is P[X1 = k]?

(b) What is the distribution of X2? That is, what is P[X2 = k]? Name the distribution of X2 and what its parameters are.

(c) Repeat the previous part for Xt for arbitrary t 1.

(d) What is the probability that at least one model drones still remains (has not crashed yet) after t days? Do not have any summations in your answer.

(e) Considering only the first day of flights, is the event A1 that the first and second model drones crash independent from the event B1 that the second and third model drones crash? Recall that two events A and B are independent if P[AB] = P[A]P[B]. Prove your answer using this definition.

(f) Considering only the first day of flights, let A2 be the event that the first model plane crashes and exactly two model planes crash in total. Let B2 be the event that the second plane crashes on the first day. What must n be equal to in terms of p such that A2 is independent from B2? Prove your answer using the definition of independence stated in the previous part.

(g) Are the random variables Xi and Xj , where i < j, independent? Recall that two random variables X and Y are independent if P[X = k1 Y = k2] = P[X = k1]P[Y = k2] for all k1 and k2. Prove your answer using this definition.