**Problem 9**Let \(\{N(t), t \in [0, \infty)\}\) be a Poisson process with rate \(\lambda\). Let \(T_1, T_2, \ldots\) be the arrival times for this process. Find\[E[T_1 + T_2 + \cdots + T_{10} | N(4) = 10].\]

Answered: Let {N(t), t = [0, ∞)} be a Poisson…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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3. Jobs arrive in a computer queue in the manner of a Poisson process with intensity λ. The central processor handles them one by one in the order of their arrival, and each has an exponentially distributed runtime with parameter , the runtimes of different jobs being independent of each other and of the arrival process. Let X (t) be the number of jobs in the system (either running or waiting) at time t, where X (0) = 0. Explain why X is a Markov chain, and write down its generator. Show that a stationary distribution exists if and only if λ < μ, and find it in this case.
x(t) == ()-1 h(t)=pt .u(t) y(t) Z(t) is a sequence of Poisson impulses. X is the expected number of impulses per Find a unit time E[Ÿ(t)] b) Ryy (T).. c) Syy (w)
2. (Stochastic Gradient Descent) Let us assume the function f(x) can be written as: ΤΙ f(x) = f(x), i=1 where fi, f2fn are L-smooth convex functions. Let i be a random variable uniformly distributed in {1, 2,..., n}. Then prove that E[||Vƒ; (x) — Vƒ; (x*)||₁₂] ≤ 2L(f(x) − f(x*)), where the expectation is taken with respect to the randomness i. Hint: You can assume that for any L-smooth convex function h the following holds: Lemma 2 (Convexity and smoothness). For all x, y € Rd ||Vh(x) - Vh(y)||≤2L(h(x)-h(y) - Vh(y) (x − y)). 1
Suppose P(X = 2) = 1/2, P(X = 5) = 1/3 and P(X = 7) = 1/6(a) Compute the probability generating function rX(t)(b) Verify that r'X(0) = P(X = 1) and rX”(0) = 2P(X = 2)(c) Compute the moment generating function of X, mX(s)(d) Verify that m'X(0) = E(X) and mX”(0) = E(X2)
2. (Stochastic Gradient Descent) Let us assume the function f(x) can be written as: f(x) = f(x), i=1 where fi, f2f are L-smooth convex functions. Let i be a random variable uniformly distributed in {1, 2,...,n}. Then prove that E[||Vf; (x) – Vfi (x*)||₁₂] ≤ 2L(f(x) − f(x*)), where the expectation is taken with respect to the randomness i. Hint: You can assume that for any L-smooth convex function h the following holds: Lemma 2 (Convexity and smoothness). For all x, y = Rd ||Vh(x) - Vh(y)||≤ 2L (h(x)-h(y) - Vh(y) (x − y)).
8 The failure of a circuit board interrupts work that utilizes a computing system, until a new board is delivered. The delivery time, Y, is uniformly distributed on the interval one to six days. The cost of a board failure and interruption includes the fixed cost co of a new board and a cost that increases proportionally to y². If C is the cost incurred, C = Co + c₂y². (a) Find the probability that the delivery time exceeds three days. (b) In terms of co and c₁, find the expected cost associated with a single failed circuit board.
Customers arrive at a facility according to a Poisson process N(t) of rate λ = 4.5 customers/hour. Each customer is admitted to the facility with probability p = 0.8. All customers, who are not admitted, leave and do not come back. Let X(t) be the number of customers admitted from time 0 to time t. Compute the following: (a) The mean value E[X(6.5)] = 23.4 (b) The probability P(X(6.5) = 23) = = (c) The conditional probability P(X(6.5) = 23 | N(6.5) = 30) = %. %. Note: For parts (b) and (c), enter your answers as percentages but do not type the percent sign.

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Let {N(t), t = [0, ∞)} be a Poisson process with rate x. Let T₁, T2, be the arrival times for this process. Find E[T₁+T₂ + + T10|N(4) = 10]. ...