Reference no: EM132377020

Questions -

Q1. Consider two machines maintained by a single machine repair robot. Machine 1 works for an exponentially distributed amount of time .with mean 3 weeks before it fails and machine 2 works for an exponentially distributed amount of time with mean 4 weeks before it fails. It takes the machine repair robot an exponentially distributed amount of time with mean 2 weeks to repair either machine. The machines are repaired in the order in which they fail. Initially, both machines are working.

(a) Formulate a continuous-time Markov chain model for the problem, with state space E = {0, 1, 2, 12, 21}, by specifying π⁽⁰⁾ and Q.

(b) Draw the corresponding transition rate diagram.

(c) What is the long run probability that both machines are under repair?

(d) How long is the machine repair robot initially expected to wait for a machine to fail?

(e) Suppose machine 2 has failed and is under repair. Find the probability that the machine repair robot fixes machine 2 before machine 1 fails as well.

(f) Suppose now that the machine repair robot itself can fail, and it works for an exponentially distributed amount of time with mean 208 weeks. When it fails, it is repaired by a robot repair robot in an exponentially distributed amount of time with mean 26 weeks. By defining an appropriate state-space and assuming that the machine repair robot is initially working, repeat (a)-(c) in this setting. For the repetition of (c), the machine repair robot also has to be working.

Q2. Customers arrive to a coffee cart according to a Poisson process with constant rate 12 per hour. Each customer is served by a single server and this takes an exponentially-distributed amount of time with mean 2 minutes irrespective of everything else. When the coffee cart opens for service, there are already 7 people waiting. Denote by X = (X_t,t ≥ 0) the number of people waiting or in service at the coffee cart t hours after it opens.

(a) Formulate a continuous-time Markov chain model for the problem for X, by specifying the state space E, π⁽⁰⁾, and Q.

(b) Draw the corresponding transition rate diagram.

(c) What is the long run probability that it customers are waiting or in service, for n = 0, 1, . . .?

(d) What is the long run expected number customers waiting or in service?

Q3. Let (Ω, F, P) be a probability space, and A₁, A₂, . . . be an increasing sequence of events; that is, A₁ ⊆ A₂ · · ·. Using only the Kolmogorov axioms, prove that P is continuous from below:

lim_n→∞P(A_n) = P(_n=1U^∞A_n).

Hint: Work with a new sequence of events B₁ := A₁ and B_n := A_n\A_n-1.

Q4. Let (Ω, F, P) be a probability space, and A₁, A₂, . . . be a decreasing sequence of events; that is, A₁ ⊇ A₂ ⊇ · · ·. Using the Kolmogorov axioms and/or the continuity from below property, prove that P is continuous from above:

lim_n→∞P(A_n) = P(_n=1?^∞A_n).

Q5. Let (Ω, F, P) be a probability space and X : Ω → R be a random variable. Denote the Borel σ-algebra on R as B(R), and define the function μ_X(B) = P(X^-1(B)) for all B ∈ B(R). Show that (R, B(R), μ_X) is a probability space. Hint: Use the fact that P is a probability measure and the definition of a random variable.

Q6. Let (Ω, F, P) be a probability space and let X : Ω → R be any simple random variable defined via

X(ω) = _i=1Σⁿb_iI_{A_i}(ω),

where n ∈ {1, 2, . . . }, A₁, . . . , A_n ∈ F b₁, . . . , b_n ∈ R, and I_{A_i} is the indicator function associated with the event A_i. Show that, without loss of generality, it can be assumed that the {A_i}_i=1ⁿ are pairwise disjoint; that is, A_i ? A_j = ∅ for i ≠ j.

7. Recall that the cumulative distribution function (cdf) of a (proper) random variable X is the function F_X : R → [0, 1] via F_X(x) := P(X ≤ x) for all x ∈ R. Using the Kolmogorov axioms and the definition of a random variable, prove:

(a) F_X is increasing; that is, x ≤ y implies F_X(x) ≤ F_X(y).

(b) F_X is right-continuous; that is lim_ε↓0 F_X(x + ε) = F_X(x). Hint: Use the continuity of P and the definition of a random variable.

Q8. There is a remarkable trigonometric identity

sin(t)/t =_n=1Π^∞cos(t/2ⁿ),

which can be viewed through the lens of probability theory, by interpreting it as an identity involving characteristic functions.

(a) Determine the distribution of the (two-point, symmetric, discrete) random variable X_n, whose characteristic function is given by cos (2^-nt).

(b) Determine the distribution of the (absolutely continuous, symmetric) random variable Y (with bounded support) whose characteristic function is given by t^-1 sin(t).

(c) Suppose X₁, X₂, . . . is a sequence of independent random variables distributed as in (a), and define Y_n = _i=1ΣⁿX_i for n = 1, 2, . . .. Show that Y_n converges to Y in distribution.