3. Let X₁, X2, Xn represent a random sample from a Rayleigh distribution with proba-bility density function given below....9Xf(x) = -=- e – x² / (20)Ө(a) Show that E(X²) = 20.Look up the gamma function, I(z) = f*9x > 0t²-¹e-t dt.(b) Find E (Σ½-1 X²) and hence find an unbiased estimator of 0.i=1(c) Estimate from the following observations using the estimator found in part (b).16.88 10.23 4.59 6.66 13.6814.23 19.87 9.40 6.51 10.95

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Answered: 3. Let X₁, X2, Xn represent a random…

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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4.) The density function of X is (2x, f(x) .0, (a) Find the mean E(3X+1); (b) Find the variance 04x+5². 0 < x < 1, elsewhere.
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(b) Let X1, X2, ...,X, be a random sample from the pdf Be z for z 2 0 f(z,0) = 0, otherwise (1) Show that the statistic T(X1, X2, . . , Xn) = X, is complete sufficient for 8. %3D i=1
2. In probability, it is common to model the deviation of a day's temperature from the monthly average temperature using the Gaussian probability density function, 1.-/9 f(t) = V9T This means that the probability that the day's temperature will be between t = a and t = b different from the monthly average temperature is given by the area under the graph of y = f(t) between t a andt = b. A related function is %3D F(x) = -2/9 dt, 2 0. %3D V9n Jo This function gives the probability that the day's temperature is between t = -x and t = x different from the monthly average temperature. For example, F(1) 0.36 indicates that there's roughly a 36% chance that the day's temperature will be within 1 degree (between 1 degree less and 1 degree more) of the monthly average. 1 (a) Find a power series representation of F(x) (write down the power series using sigma notation). (b) Use your answer to (a) to find a series equal to the probability that the day's temperature will be within 2 degrees of the…
Develop two Monte Carlo integration approaches to estimate x exp(-x)dx. (You must use different distributions in the two approaches). Check your answer using the integrate() function.
Suppose the life span of a laboratory animal is defined by a Rayleigh density function with a=0 and b=30 weeks: (x-a)² bu(x-a) Where £x (x)= = b (x-a)e 9 = { 1 u(x) = x>0 x < 0 What is the probability that the animal will live between 10 and 20 weeks? What is the expected lifespan of the animal?
33.-
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Suppose that each individual in a large insurance portfolio incurs losses according to an exponential distribution with mean 1/2,where i varies over the portfolio according to a G(a, 3) mixing distribution. The respective densities of the two distributions are given by Sx (x) = (1/2) exp (-x/à), x > 0, 2 > 0; S(2) =- T(a) 2-l exp(-8i), 2 > 0. Given that the Pareto pdf given by fx (x) = (5 +x)ª+1 * > 0, a > 0, 8> 0. (a) Show that the marginal distribution of losses follows a Pareto distribution, i.e. P(a, 5). (b) Use the mixing formulation of the Pareto to deduce that if X~P(a, 5), then E (X) = .

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