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Q. 2 a) Using only Unif 0,1) random variates, use a Monte Carlo algorithm to approximate...
Q. 2 a) Using only Unif 0,1) random variates, use a Monte Carlo algorithm to approximate the value of the Gamma function 0 by considering the function as an expectation of a function of a random variable. b) Show that if a Monte Carlo simulation of size N is used then the variance of the Monte Carlo estimator is Var (「(a))- 2a-1)-[「(a)]2] provided that α > 0.5. c) Write an R function to implement the method returning the estimated value...
1. (10 marks) random variable with density r(x). Let g: R - (a) Let X R...
1. (10 marks) random variable with density r(x). Let g: R - (a) Let X R be a (differentiable) function and let Y = g(X). Write expressions for the following ((ii)-(iv) should be in terms of the density of X (i) The integral f()d (ii) The mean E(X) (ii The probability P(X e (a, b) (iv) The mean E(g(X)) R be a smooth (1 mark (1 mark) (1 mark (1 mark) (b) Let z E R be a constant and...
Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y :=...
Let X1 d= R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of Y...
Let Y-ar+b (a) Find the mean and variance of Y in terms of the mean and variance of X b) Evaluate...
Let Y-ar+b (a) Find the mean and variance of Y in terms of the mean and variance of X b) Evaluate the mean and variance ofY if Xhas the following PDF: (a)-ele (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance d) Evaluate the mean and variance of Yif X-bcos 2U) where U is a uniform random variable in of 1 the unit interval.
Let Y-ar+b (a) Find the mean...
SOLVE the following in R code: iid Let X1, , Xn ~ U (0,0). We are...
SOLVE the following in R code:
iid Let X1, , Xn ~ U (0,0). We are going to compare two estimators for θ: 01-2X, the method of moments estimator -maxX.... X1, the maximum likelihood estimator I. Generate 50,000 samples of size n-50 from U(0,5). For each sample compute both θ1 and 02 (Hint: You can use the R cornmand max (v) to find the maximum entry of a vector v). The results should be collected in two vectors of length...
Let X1 d = R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to...
Let X1 d = R(0,1) and X2 d= Bernoulli(1/3) be two independent random variables, define Y := X1 + X2 and U := X1X2. (a) Find the state space of Y and derive the cdf FY and pdf fY of Y . (You may wish to use {X2 = i}, i = 0,1, as a partition and apply the total probability formula.) (b) Compute the mean and variance of Y in two different ways, one is through the pdf of...
1. Let p(x), a(x) and B(x) be three functions of r. Consider the PDE of u(r, t): PEx) at (a) (10 ...
1. Let p(x), a(x) and B(x) be three functions of r. Consider the PDE of u(r, t): PEx) at (a) (10 pt) Show that the method of separation of variables works only if ρ(r) equals to a constant. (b) (10 pt) Assumefor some constant c. Show that the spatial equation (the differential equation about the spatial variable r) is of Sturm-Liouville type. B(z)
1. Let p(x), a(x) and B(x) be three functions of r. Consider the PDE of u(r, t):...
Problem1 Let Y=aX + b . (a) Find the mean and variance of Y in terms of the mean and variance of ...
Problem1 Let Y=aX + b . (a) Find the mean and variance of Y in terms of the mean and variance of X (b) Evaluate the mean and variance ofYifXhas the following PDF (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance of 1 d) Evaluate the mean and variance of Yif X bcos(2RU) where U is a uniform random variable in the unit interval.
Problem1 Let Y=aX + b...
work step by step. Thanks الم 3. Let k : (0,1] x [0, 1] + R...
work step by step. Thanks
الم 3. Let k : (0,1] x [0, 1] + R be a continuous function and let f be a Lebesgue integrable function on (0,1). (a) Show that for each y € (0,1), 2 + f(-x){}(2", y) is Lebesgue integrable on (0,1). (b) Define g : [0, 1] +R by 8(u) = Sam Slam)x(x, y)dır. 10,11 Prove that g is continuous at cach y € (0, 1].
[20 points] Problem 2 - Monte Carlo Estimation of Definite Integrals One really cool application of...
[20 points] Problem 2 - Monte Carlo Estimation of Definite Integrals One really cool application of random variables is using them to approximate integrals/area under a curve. This method of approximating integrals is used frequently in computational science to approximate really difficult integrals that we never want to do by hand. In this exercise you'll figure out how we can do this in practice and test your method on a relatively simple integral. Part A. Let X be a random...
Q. 2 a) Using only Unif 0,1) random variates, use a Monte Carlo algorithm to approximate the value of the Gamma function 0 by considering the function as an expectation of a function of a random variable. b) Show that if a Monte Carlo simulation of size N is used then the variance of the Monte Carlo estimator is Var (「(a))- 2a-1)-[「(a)]2] provided that α > 0.5. c) Write an R function to implement the method returning the estimated value...
1. (10 marks) random variable with density r(x). Let g: R - (a) Let X R be a (differentiable) function and let Y = g(X). Write expressions for the following ((ii)-(iv) should be in terms of the density of X (i) The integral f()d (ii) The mean E(X) (ii The probability P(X e (a, b) (iv) The mean E(g(X)) R be a smooth (1 mark (1 mark) (1 mark (1 mark) (b) Let z E R be a constant and...
Let Y-ar+b (a) Find the mean and variance of Y in terms of the mean and variance of X b) Evaluate the mean and variance ofY if Xhas the following PDF: (a)-ele (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance d) Evaluate the mean and variance of Yif X-bcos 2U) where U is a uniform random variable in of 1 the unit interval.
Let Y-ar+b (a) Find the mean...
SOLVE the following in R code:
iid Let X1, , Xn ~ U (0,0). We are going to compare two estimators for θ: 01-2X, the method of moments estimator -maxX.... X1, the maximum likelihood estimator I. Generate 50,000 samples of size n-50 from U(0,5). For each sample compute both θ1 and 02 (Hint: You can use the R cornmand max (v) to find the maximum entry of a vector v). The results should be collected in two vectors of length...
1. Let p(x), a(x) and B(x) be three functions of r. Consider the PDE of u(r, t): PEx) at (a) (10 pt) Show that the method of separation of variables works only if ρ(r) equals to a constant. (b) (10 pt) Assumefor some constant c. Show that the spatial equation (the differential equation about the spatial variable r) is of Sturm-Liouville type. B(z)
1. Let p(x), a(x) and B(x) be three functions of r. Consider the PDE of u(r, t):...
Problem1 Let Y=aX + b . (a) Find the mean and variance of Y in terms of the mean and variance of X (b) Evaluate the mean and variance ofYifXhas the following PDF (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance of 1 d) Evaluate the mean and variance of Yif X bcos(2RU) where U is a uniform random variable in the unit interval.
Problem1 Let Y=aX + b...
work step by step. Thanks
الم 3. Let k : (0,1] x [0, 1] + R be a continuous function and let f be a Lebesgue integrable function on (0,1). (a) Show that for each y € (0,1), 2 + f(-x){}(2", y) is Lebesgue integrable on (0,1). (b) Define g : [0, 1] +R by 8(u) = Sam Slam)x(x, y)dır. 10,11 Prove that g is continuous at cach y € (0, 1].
[20 points] Problem 2 - Monte Carlo Estimation of Definite Integrals One really cool application of random variables is using them to approximate integrals/area under a curve. This method of approximating integrals is used frequently in computational science to approximate really difficult integrals that we never want to do by hand. In this exercise you'll figure out how we can do this in practice and test your method on a relatively simple integral. Part A. Let X be a random...
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