2. Let X = 8 + Xt-1 + Wt-1, for t=1,2,...., with Xo=0, where {W}~WN(0,0²). Note whenthe drift term (8) is zero, X is a random walk. We can also write Xt = St +Σ=1 W₁, t = 1,2 ...a) Derive ux (t) and yx (t + h, t)b)Is {X} stationary? Why/Why not?

To fully understand and solve the problem related to the stochastic process Xt = δ + Xt-1 + Wt-1,…

Answered: 2. Let X = 8 + Xt-1 + Wt-1, for…

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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2.5.7. Let the random variables X1 and X2 have the joint pdf f(x1,x2) = 1/T, for (x1 – 1)2 + (2 +2)2 < 1, zero elsewhere. Find fi(æ1) and f2(x2). Are X1 and X2 independent? %3D
11. Let (yı, Y2, .., Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that –2) In Y; Xản i=1
Imagine you have two estimators X and Y, for unknown parameter µ. Assume that they are estimated from independent datasets. Let Z = }(X+Y) be the average of these two estimators. To write the symbol µ, you can write mu. For other math notation, you can E[X], Var(X) and (X+Y)/2 or (1/2)(X+Y). Part a - If X and Y are unbiased estimators, E[X] = E[Y] = µ, then is Z unbiased? Explain why or why not. Your explanation should include a short derivation. Part bL Assume Var(X) = Var(Y). Is Z a lower variance or higher variance estimator than X? Explain your answer. Your explanation should include a short derivation.
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Q. 5 Let X be a continuous uniform random variable on [-a, a]. Find the pdf for Y = g(X), where 0, x b. g(x) = 3x,
Q7. Let X; (i = 1,.,n) be a random sample from the N(u, o²) population with unknown parameters u and o?. Then we know that T, = E-1(X - X)²/(n – 1) is an unbiased estimator -xi-1 . Now consider two other estimators T, and T3 for o? where of a? and further, (n-1)T T2 = E(X – X)*/n and T3 = E(X – X)²/n + 1). %3D What is the variance of T,? Find out the variances and biases of T2 and T3 and thus obtain their m.s.e.'s. Show that among the three estimators T,, T2 and T3 the one with minimum m.s.e. is T3.
Assume b, c are independent random variables with uniform distribution on the interval [-L, L), where L e R'. Find the probability that the solutions of the quadratic equation x? + bx + c= 0 are real. What happens when L → +0?
2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.
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Suppose claim amounts at a health insurance company are independent of one another. In the first year calim amounts are modeled by a gamma random variable X with alpha=40, and beta=3. In the second year, individual claim amounts are modeled by random variable Y=1.05X+3. Let W be the average of 30 claim amounts in year two set up the equation to model the random variable W. a) Find the moment generating function of W b) Based on moment generating function of W is W also a gamma distribution? if so what are the parameters? c) Find the approximate probability that W is between 125$ and 130$.
The moment generating function M (t) for random variables y are as follow. Examine and identify the distribution of the random variables and state the parameter involves. 1. M(t) = e +2 3 1 2. M()-(3-28) = 3. M(t)= -(-) 4. M(t)= 5 5-t
Only part E

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2. Let X = 8 + Xt-1 + Wt-1, for t=1,2,...., with Xo=0, where {W}~WN(0,0²). Note when the drift term (8) is zero, X is a random walk. We can also write Xt = St + Σ₁=1 W₁, t = 1,2 ... a) b) Derive µx (t) and yx (t + h, t) Is {X} stationary? Why/Why not?