QUESTION 2 A poisonous gas sensor measures a continuous random variable X. In the presence of...
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample...
(9) [12 pts] An exponentially distributed random variable, call it X, has the following probability density functior f(x)-oe ex , x > 0, θ > 0 Note that ElX] and VX]ー1 For the rest of this question, assume that you have a data set (xn1 consisting of a random sample of N observations of X. (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that...
2 Let X1, X2, ...,X, be independent continuous random variables from the following distribution: f(3) = ox-(0-1) where : > 1 and a > 1 You may use the fact: E[X]- .- 2.1 Show that the maximum likelihood estimator of a is ômle = Ei log Xi 2.3 Derive a sufficient statistic for a. What theorem are you using to determine sufficiency? 2.4 Show that the fisher information in the whole sample is: 1(a)= 2.5 What Cramer Rao lower bound...
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...