. Continuation. In the context of the preceding exercise, a typical system initially operates for a time Y = min{X1, X2} with two components and then operates for a time Z = max{X 1 , X 2 } − Y with...

. Continuation. In the context of the preceding exercise, a typical system initially operates for a time Y = min{X1, X2} with two components and then operates for a time Z = max{X₁, X₂} − Y with one component. Thereupon it fails. Find the distributions and means of Y and Z. Find the distribution of Z conditioned that X₁
> X₂. You might want to use the memory less property of the exponential distribution in Exercise 3. Find the distribution of Z conditioned that X₂
> XExercise 3Bernoulli Process. Consider a sequence of independent Bernoulli trials in which each trial results in a success or failure with respective probabilities p and q = 1 − p. Let N(t) denote the number of successes in t trials, where t is an integer. Show that N(t) is a discrete-time renewal process, called a Bernoulli Process. (The parameter t may denote discrete-time or any integer referring to sequential information.) Justify that the inter-renewal times have the geometric distribution P{ξ1 = n} = pq_n−1, n ≥ 1. Find the distribution and mean of N(t), and do the same for the renewal time T_n. Show that the moment generating function of T_n
is