6. In a discrete time input-output system, let Xn denote the number of units of an...

6. In a discrete time input-output system, let Xn denote the number of units of an item at end of the nth period, (n − 1, n]. In each period (n − 1, n], the system has an input(arrival) Vn and an output (departure) Un, n ≥ 1. The system disregard any output if its exit makes the state of the system negative. We assume that {(Un, Vn)|n ≥ 1} is an i.i.d. sequence such that Un is independent of Vn for all n and P{U1 = k} = qk, k ≥ 0 and P{V1 = k} = pk, k ≥ 0. Assume that X0 is Z +-valued random variable independent of {(Un, Vn)|n ≥ 1}. Show that {Xn|n ≥ 0} is a Markov chain.

Q7. Let {Xn|n ≥ 0} be as in Q6. Show that Xn+1 = X0 + Zn+1 − Xn m=0 min{0, Xm + Vm+1 − Um+1}, where Zn = Xn m=1 (Vm − Um), n ≥ 1.

Q8. Let {Xn|n ≥ 0} be as in Q7. Show that Xn = max{X0 + Zn, max 1≤m≤n Zn − Zm}, n ≥ 1

Expert Solution

orchestra answered 1 year ago

Let V denote the number of units of a variable input (i.e., nitrogen fertilizer) that is...

Let V denote the number of units of a variable input (i.e., nitrogen fertilizer) that is used in combination with a fixed input (i.e., land). Let TP denote the total amount of production of a crop (i.e., corn) that is obtained from using each input level. Point A is a point of inflection. 1. TP increases at a decreasing rate ____________. (Points: 20) a. from O to A b. from A to C c. beyond point O d. beyond point C...

a die is rolled 6 times let X denote the number of 2's that appear on...

a die is rolled 6 times let X denote the number of 2's that appear on the die. 1. show that X is binomial. 2. what is the porbaility of getting at least one 2. 3. find the mean and the standard deviaion of X

Q1. Let {Xn : n ≥ 0} denote the random walk on 9-cycle. Express it as...

Q1. Let {Xn : n ≥ 0} denote the random walk on 9-cycle. Express it as a random walk on a group (G, ·) with transition probabilities given by pxy = µ(y · x −1 ) for an appropriate distribution µ on G. Q2. Consider the stochastic process {Xn|n ≥ 0}given by X0 = 1, Xn+1 = I{Xn = 1}Un+1 + I{Xn 6= 1}Vn+1, n ≥ 0, where {(Un, Vn)|n ≥ 1} is an i.i.d. sequence of random variables such...

Consider the discrete-time LTI system characterized by the following difference equation with input and initial conditions specified

Consider the discrete-time LTI system characterized by the following difference equation with input and initial conditions specified: y[n] - 2 y[n-1] – 3 y[n-2] = x[n] , with y[0] = -1 and y[1] = 0, x[n] = (-1/2)n u[n-2]. ? Write a MATLAB program to simulate this difference equation. You may try the commands ‘filter’ or ‘filtic’ or create a loop to compute the values recursively. ? Printout and plot the values of the input signal, x[n] and the...

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum...

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum of the positive divisors of n (as in the notes). (a) Evaluate τ (1500) and σ(8!). (b) Verify that τ (n) = τ (n + 1) = τ (n + 2) = τ (n + 3) holds for n = 3655 and 4503. (c) When n = 14, n = 206 and n = 957, show that σ(n) = σ(n + 1).

There is a box with space for 16 items. Let A denote the number of things...

There is a box with space for 16 items. Let A denote the number of things that are type one and B the number of things that are type two. Assume that A and B are independent random variables. Assume that all possible (a,b) pairs are equally likely. I) How many possible pairs (a,b) are there? II) Which event is more likely {A = 1} or {B = 0}? Justify your answer. III) Compute P(B=5) and P(A=10) IV) If there...

Each time draw 6 different numbers from 1 ~ 45 randomly Let M denote the times...

Each time draw 6 different numbers from 1 ~ 45 randomly Let M denote the times you have to draw such that all the number from 1~45 has been drawn What's the expected number of M. please explain thoroughly. Thanks.

5. Calculate the molar mass of barium hydroxide. (Input only the number, not the units.) 6....

5. Calculate the molar mass of barium hydroxide. (Input only the number, not the units.) 6. What is the mass percent of oxygen in phosphoric acid, H3PO4? (Input just the number, not the percent sign. Use correct sig. figs. from the periodic table at the start of the exam.)

In a sequence of independent flips of a fair coin, let N denote the number of...

In a sequence of independent flips of a fair coin, let N denote the number of flips until there is a run of three consecutive heads. Find P(N ≤ 8). (Should write out transition matrix.)

Flip a fair coin 4 times. Let ? and ? denote the number of heads and...

Flip a fair coin 4 times. Let ? and ? denote the number of heads and tails correspondingly. (a) What is the distribution of ?? What is the distribution of ? ? (b) Find the joint PMF. Are ? and ? independent? (c) Calculate ?(? ?) and ?(X≠?)(d) Calculate C??(?, ? ) and C???(?, ? )

Question