Dennis is testing model airplanes. He starts with n model planes which each independently have probability p of flying successfully each time they are flown, where 0 (a) What is the distribution of X? That is, what is PX,-
(b) What is the distribution of Xs? That is, what is PK2 Show that X2 follows a binomial distribution by finding some n' and p' such that X2Binom(n',p')
(c) Repeat the previous part for X, for arbitrary t 21.
(d) What is the probability that at least one model plane 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 planes crash independent from the event B1 that the second and third model planes crash? Recall that two events A and B are independent if PAnB PAPB. 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 X, and Xj, where i < j, independent? Recall that two random vari ables X and Y are independent if PX = ki n Y = k2] = PlX = kJEY Prove your answer using this definition k2] for all ki and k2.