2. A distribution often used to model the lifetime of electronic components is the Rayleigh density...
QUESTION 1 The length of life of an electronic component used in a guidance control system for missiles is assumed to follow a Weibull distribution with density given by >0, θ > 0 Let Yı,Y2, ,Y10 denote a random variables for the lifetime of a sample of size n= 10 of these electronic components. We wish to construct a 95% confidence interval for θ (a) Find the maximum likelihood estimator, 6, of 0. (b) Find the distribution of U =...
The Rayleigh density function is given by 2y) -y2 е ө y >0 f(y) = --{@ elsewhere The quantity Y? has an exponential distribution with mean o. If Yı, Y2, ..., Yn denotes a random sample from a Rayleigh distribution, show that Wn = ?=1 Y/? is a consistent estimator for e.
4. Let y1θ ~iid Uniform (0,0), for i-1, n, Assume the prior distribution for θ to , be Pareto(a, b), where p()b1 for 0> a and 0 otherwise. Find the posterior distribution of θ.
3. The Rayleigh distribution is a continuous distribution with pdf of the form Så exp(-+) $(30) = >0 otherwise Suppose that X1,..., X, form a random sample from a Rayleigh distribution where the value of the parameter 8 >0 is unknown. a. Find the maximum likelihood estimator (MLE) of e, assuming that all observed values satisfy 2: >0. b. Is your MLE of 8 a sufficient statistic? Why or why not?
Problem 7: 10 points Assume that a lifetime random variable (T) is exponentially distributed with the intensity λ > 0. I. Determine conditional density of the residual lifetime, T-u, given that T 〉 u. 2. Find conditional expectation, E TT>u
Written Problems l. Let Yı, Ya, Ya be a random sample from an Exponential distribution with density function f(y)-Te-3, y > 0. Find the MSE of each of the following estimators of θ: (a)-互华 (c) θ=F 2
6. Suppose that X and Y are jointly continuous random variables with joint density f(r, y)otherwise (a) Given that X > 1, what is the expected value of Y? That is, calculate Ey X 〉 1).
Example 7. Let Y1, ... ,Yn be a random sample from a Rayleigh distribution with pdf Ske-?/(20) f(y\C) = 10 = if y>0,0 > 0 otherwise otherwise Find a sufficient statistic for 0.
A certain type of electronic component has a lifetime Y (in hours) with probability density function given by That is, Y has a gamma distribution with parameters α = 2 and θ. Let denote the MLE of θ. Suppose that three such components, tested independently, had lifetimes of 120, 130, and 128 hours. a Find the MLE of θ. b Find E() and V(). c Suppose that actually equals 130. Give an approximate bound that you might expect for the error of estimation. d What...
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...