20. Suppose {Xt, t≥ 0) is a continuous time stochastic process on the proba-bility space (S2, B, P) whose paths are continuous. We can understand thisto mean that for each t≥ 0, X,R is a random variable and, foreach we, the function t → X,(w) is continuous; that is a member ofC[0, ∞o). Let T: 2 [0, ∞o) be a random variable and define the processstopped at 7 as the function X, [0, ∞o) defined byX₁ (w):= Xr(w) (w), WES.Prove X, is a random variable.

Given::- Suppose \{X_{t}, t >= 0\} is a continuous time stochastic process on the proba- bility…

Answered: : 20. Suppose (Xt, t≥ 0} is a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step t> 1 a new node is added to the network. The node arrives together with m = 2 new links, which are connected to m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: ki II¿ = Z - 1 N(t-1) with Z(k-1) - j=1 where k, is the degree of node i, and N(t 1) is the number of nodes in the network at time t-1. (b) What is the average node degree (k) at time t? What is the average node degree in the limit t→ ∞o?
J.UZ! b) Let X₁, X₂,..., X and Y₁, Y₂,, Ym be random samples from populations with moment generating 25 functions Mx, (t) = e3t+t² and My, (t) = , respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ - X3 + x4 + X5. ii) What is the value of the sample size n, if P[Σ(X₁-X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Y-µy ≥10) < 0.04?
Prove the following
3. If the moment-generating function of a random variable X is Mx (t) = (1-5 (a) Find E(X). (b) Find a mode. 1x (0) = (4-5()-3) X Gamica (515)
Prove the theorem
b) Let X₁, X₂, ..., Xn and Y₁, Y₂, ..., Ym be random samples from populations with moment generating functions Mx₁(t) = e³t+t² and My(t) = (₁)², respectively. iii) What is the value of the sample size m, if P(|Ỹ − µỹ| ≥ 10) < 0.04?
a) Let X₁, X2,..., X be a random sample of size n from population X. Suppose that X has the following probability mass function: (0(1-0)*-1, (o, i) Find the conditional distribution of the likelihood function given the function of the statistic, 6 = ₁X₁. Hint: Apply f(L(0)|g(0,0)). ii) Based on i) above, is = 1X₁ a sufficient statistic? Justify your answer. B ii) Show whether or not X, is a consistent estimator of the parameter f(x; 0) = x = 1,2,3,... elsewhere
2. In a different warehouse, book orders are handled by two members of staff, A and B, who take turns to pack each orders as it arrives (this is a non-random alternation). Assume orders arrive as a Poisson process with rate A per hour, that A packs the first order, and that orders are packed instantaneously. Let NA(t) be the number of orders packed by A by time t, and let X₁, X2.... be the inter-arrival times for the process (NA()). (a) What is the probability distribution for X.? Hint: relate the X, to the times between orders. (b) Is (NA(t)) a Poisson process? Explain your reasoning.
3. Consider the linear probability model Y₁ = Bo+B₁Xi+ui, where Pr(Y₁ = 1|Xi) = Bo +3₁ Xi. (a) Show that E(u₂|X₂) = 0. (b) Show that Var(u₁|X;) = (Bo + B₁X;)[1 − (Bo + B₁X;)]. (c) Is ui conditionally heteroskedastic? Is u heteroskedastic? (d) Derive the likelihood function.
21. Let X1, X2, ... be independent, U (0, 1)-distributed random variables, and let Nm E Po(m) be independent of X1, X2, Set V= max{X1,..., (Vm = 0 when Nm =0). Determine (a) the distribution function of Vm, (b) the moment generating function of Vm. It is reasonable to believe that V is "close" to 1 when m is "large" (cf. Problem 8.1). The purpose of parts (c) and (d) is to show how this can be made more precise. (c) Show that E V, (d) Show that m(1 - V) converges im distribution as m → x. and de- termine the limit distribution. →l as m→.
Prove the following
Find the PSD of a random process X(t) if E[X(t)] =1 and R(T) = 1 + e-aIt, X,

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