Kindly answer the questions. Many thanks

Reading: Work through Pitman 4.1 and 4.2, particularly attending to the diagrams on pp. 262-263 illustrating probabilities as areas under a curve for discrete and continuous distributions. Problem 1 (Pitman 3.4 #15: Memoryless property of geometric distribution). Suppose that F has geometric distribution on {0, 1, 2, . . .}. (a) Show that for every k 0, P (F k = m|F k) = P (F = m), m = 0, 1, 2, . . . (b) Show that the geometric distribution is the only discrete distribution on {0, 1, 2, . . .} with this property. Problem 2. Let X be a random variable with density ( ce2x if x 0 f (x) = 0 otherwise, where c is a constant. (a) Find c. (b) Compute E(X). (c) Compute V ar(X). For full credit you must show the full computations for all parts of this problem. Problem 3. Let Z = (X1 , X2 , X3 ) be a random point, written in rectangular coordinates X1 , X2 , X3 , chosen from the uniform distribution on the interior of the unit ball in 3 dimensions. p (a) What is the probability density of the distance R = X12 + X22 + X32 ? (b) What is the probability density of the first coordinate X1 ? Problem 4 (Pitman 4.2 #4 + (f)). Suppose component lifetimes are exponentially distributed with mean 10 hours. Find: (a) the probability that a component survives 20 hours; (b) the median component lifetime; (c) the SD of component lifetime; (d) the probability that the average lifetime of 100 independent components exceeds 11 hours; (e) the probability that the average lifetime of 2 independent components exceeds 11 hours. 1 (f) the probability that the minimum lifetime of 2 independent components exceeds 11 hours. Problem 5. Buses arrive at 116th and Broadway at the times of a Poisson arrival process with intensity arrivals per hour. These may either be M104 buses or M6 buses; the chance that a bus is an M104 is 0.6, while the chance that it is an M6 is 0.4, and the types (M6 or M104) of successive buses are independent. (a) If I wait for an M104 bus, what is the chance that I will wait longer than x hours? (b) What is the probability that two M6 buses and no M104 buses arrive in the first x hours? (c) What is the expected number of hours until the third M6 arrives? (d) What is the variance of the number of hours until the third M6 arrives