1. Suppose that {Yin: 1 ≤ i ≤n; n = 1,2,...} is a triangular array of random variables such thatfor each n, Yl,n..., Yn,n are independent and identically distributed;√n E(Y₁.n) → (with ||

Convergence in probability: A sequence of random variables Xn is said to be convergence in…

Answered: 1. Suppose that {Yin: 1 ≤ i ≤n; n =…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

Let X Exponential (3) a) Find the Markov upper bound for P(X>10). b) Find the Chebyshev upper bound for P(X>10). State which of the two bounds found is better. c) Find the Chernoff upper bound for P(X>10). Clearly state the allowable range for t. d) Let Zn be a RV from a Gamma(n, 3) distribution for n=1,2,... . What does converge in probability to? Please state both the value and the reasoning by which Z converges in probability to it. n
6. Let X₁, X2, ..., Xn be a random sample on the random variable that is uniformly distributed over the interval (1,0) (a) Find the moment estimator, 0* of and the maximum likelihood estimator 0 of 0 (b) Show that 0* is unbiased but Ô is biased (c) Compute the bias of Ô and find the estimator ♬ that is a function of ô and unbiased. (d) Find the variance of 0* and 7 (e) Show that both * and are consistent estimators of (f) Find RE (7,0*) and comment on your results
We say that a mean-zero random variable X is sub-Gaussian with parameter o² if the m.g.f. of X satisfies, for all t ER, Mx(t) ≤e ²0¹2 Let X be a sub-Gaussian random variable with parameter o². Using Markov's inequality, show that, for all a > 0, P(X ≥ a) ≤ e¯20² (Hint: the method of proof of the one-sided Chebyshev inequality may provide some in- spiration.)
None
20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 2
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.
Let X be a Markov chain on S, and let I: S" {0, 1}. Show that the distribution of Xn, Xn+1,..., conditional on {I (X₁,..., Xn) = 1} n {Xn = i}, is identical to the distribution of Xn, Xn+1,... conditional on {Xn = i}.
19.5)
3. Let Xi ∼ Poi(1), i = 1, . . . , n, be n = 20 independent Poisson r.v’s.Hint: you may use E(Y ) = λ and Var(Y ) = λ, for a Poisson r.v. Y ∼ Poi(λ). 3a. Use the Markov inequality to obtain a bound on Pr (Ln Xi > 30) (±0.05 is okay). 3b. Use the CLT to approximate the same probability (±0.01 is okay)
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→.
3
2. Let Zbe a Markov chain on a state space S = {1,… … , m} with transition probability matrix P(i, j) = Tij. Suppose that sample paths of Z are observed from which we record Njj #i →→ j transitions of Z, m N₁ = [N₁; (# transitions out of state i). j=1 It is known that the likelihood function of the observations of Z is given by m Nij ΠΠΠ. i=1_ j=1 L = Incorporating the constraint j = 1 for each i = S, j=1 m (a) Write the loglikelihood function of the observation. (b) Show that the maximum likelihood estimator of i → j transition probability ij is N = Tij = (c) Let m = 2 and assume that the number of transitions of Z is given by the matrix Nij N₁ 50 100 25 75 Use the above information to derive estimators for 11, 12, 721 and 22.

SEE MORE QUESTIONS

Recommended textbooks for you

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

Elementary Statistics: Picturing the World (7th E…

Statistics

ISBN:

9780134683416

Author:

Ron Larson, Betsy Farber

Publisher:

PEARSON

The Basic Practice of Statistics

Statistics

ISBN:

9781319042578

Author:

David S. Moore, William I. Notz, Michael A. Fligner

Publisher:

W. H. Freeman

Introduction to the Practice of Statistics

Statistics

ISBN:

9781319013387

Author:

David S. Moore, George P. McCabe, Bruce A. Craig

Publisher:

W. H. Freeman

SEE MORE TEXTBOOKS