hi please help with detailed answers thank you lots

Question 15 0 / 3 pts Consider a Poisson process with rate 1 = 1 / hour. What is the probability that the time between the 5th and 6th arrivals is greater than 1 hour? O a. 0 orrect Answer O b. 0.368 ou Answered O C. 0.632 O d. 0.865 O e. 1Question 16 3/3 pts Consider again a Poisson process with rate 1 = 1 / hour. What is the probability that we observe exactly 1 arrival by the end of the first hour? O a. 0 Correct! O b. 0.368 O c. 0.632 d. 0.865 O e. 1Question 17 3 / 3 pts Suppose X1, . . ., Xn are i.i.d. from a Pois(2) distribution. What is the approximate distribution of the sample mean X for n = 100? O a. Pois(2) O b. Pois(0.02) O c. Nor(2, 2) Correct! O d. Nor(2, 0.02) O e. Nor(0.02, 0.02)Question 18 0 / 3 pts Suppose U and V are i.i.d. Unif(0, 1) random variables. Select from the options below an appropriate expression to generate one realization of an n-sided die toss. (Recall that [x ] is the "ceiling" or "round up" function.) O a. [nu] + [nV] O b. 0.5 x [nu] + 0.5 x [nV] orrect Answer O c. [nu] O d. All of the above. ou Answered O e. Only (b) and (c).Question 19 3 / 3 pts If U ~ Unif(0, 1), what is the distribution of -2 en(U4)? Correct! a. Exp(1/8) O b. Exp (8) O c. Erlang4 (1/2) O d. Erlang4 (2) O e. None of the above.Question 20 3 / 3 pts Suppose X has the Nor(0, 1) distribution with c.d.f. (x). Further suppose that U is a Unif(0, 1) number. What is the distribution of 20-'(U) - 1? O a. Unif(-1, 1) O b. Unif(-1, 4) O c. Nor(1, 4) O d. Nor(-1, 2) Correct! O e. Nor(-1, 4)