Homework Answers
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].
and let X and S be sample mean be a random sample from N(u,0) 1. Let...
and let X and S be sample mean be a random sample from N(u,0) 1. Let are independent, follow the and sample variance, respectively. In order to show that X and S steps below X x-x2 , and show the joint pdf of 1-1) Use the change of variable technique X,X,,X n is (n 1s 202 1 f(F,x,) = n exp 202 a27 [Hint 1] Use Jacobian for n x n variable transformation [Hint 2] 4AT-r- des dis Je ddi...
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].
and let X and S be sample mean be a random sample from N(u,0) 1. Let are independent, follow the and sample variance, respectively. In order to show that X and S steps below X x-x2 , and show the joint pdf of 1-1) Use the change of variable technique X,X,,X n is (n 1s 202 1 f(F,x,) = n exp 202 a27 [Hint 1] Use Jacobian for n x n variable transformation [Hint 2] 4AT-r- des dis Je ddi...
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