The Rayleigh density function is given by 2y) -y2 е ө y >0 f(y) = --{@...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
Problem 4 Let Yı, Y2, ..., Y, denote a random sample from the probability density function (0 + 1)ye f(0) = 0 <y <1,0 > -1 elsewhere Find the MLE for .
Let Yı, Y, have the joint density S 2, 0 < y2 <yi <1 f(y1, y2) = 0, elsewhere. Use the method of transformation to derive the joint density function for U1 = Y/Y2,U2 = Y2, and then derive the marginal density of U1.
(1 point) Let Yı, Y2, ..., Yn be a random sample from the probability density function f(yla) = |aya-2/5° f(y ) 0 <y< 5 otherwise 0 for > -1. Find an estimator for a using the method of moments.
Let X1, X2, ..., Xn be iid random variables with a "Rayleigh” density having the following pdf: 22 -12 10 f(x) = e x > 0 > 0 0 пе a) (3 points) Find a sufficient estimator for 0 using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = V c) (7 points) What is the MLE of 02 + 0 – 10 ? d) (7 points) For a fact, 21–1...
2. A distribution often used to model the lifetime of electronic components is the Rayleigh density 0 otherwise where θ > 0. (a) If Y has the Rayleigh density, find the distribution for U-Y2 (b) Use the result of part a) to find EY
2. Let Yı, ..., Yn be a random sample from an Exponential distribution with density function e-, y > 0. Let Y(1) minimum(Yi, , Yn). (a) Find the CDE of Y) b) Find the PDF of Y (c) Is θ-Yu) is an unbiased estimator of θ? Show your work. (d) what modification can be made to θ so it's unbiased? Explain.
Let Y be a random variable with probability density function, pdf, f(y) = 2e-2y, y > 0. Determine f (U), the pdf of U = VY.
1. For continuous random variables Y and Y2 with joint probability density 12/27 f02.2) = ce 0,293 Y: > 0,72 > 0 0 elsewhere i). Find Var(Y). j) Find Var(2) k) Find Cov(Y.Y). 1) Find PY,.;, that is, the correlation between Y, and Y. m) Find Var(2Y, + 372).
Let Yı, ..., Yn be an independent and identically distribution sample from the distribution function f(y) = 3y?, for 0 Sy <1. (a) Show the sample mean, y converges in probability to some constant, c. Find c. (b) Find a function that converges in probability to log(C).