Let X1, X2, X3......Xn represent a random sample from the Rayleigh distribution with destiny function given exercise 15. Determine:

a) The maximum likelihood estimator of theta and then calculate the estimate the vibratory stress data gven in the exercise. Is this estimator the same as the unbiased estimator suggested in Exercise 15.

b. The mle of the median of the vibratory stress distribution. (Hint: First express the median in the terms of theta.)

15. Let X1, X2, X3......Xn represent a random sample from the Rayleigh distribution with PDF

f(x;theta) - (X/theta)e^-x^2/2(theta) X>0

a) It can be shown that E(X^2) = 2(theta). Use this fact to construct an unbiased estimator of theta based on (see document) (and use rules of expected value to show that it is unbiased).

b. Estimate (theta) from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions:

Question Details:
Let X1, X2, ...., Xn be uniformly distributed on the interval (0 to a). Recall that the maximum likelihood estimator of a is â =max(Xi).
a) Argue intuitively why â cannot be an unbiasedestimator for a.
b) Suppose that E(â ) = na /(n+1).
Is it reasonable that â consistently underestimates a

The asking problem:
The uniform rod AB lies in a vertical plane. Its ends are connected to rollers which rest against frictionless surfaces. Determine the relation between the angles theta and alpha when the rod is in equilibrium.
Note: This is a 2D problem and not a 3D. It was taken from the «Statics of Rigid Bodies in Two D

#1
Suppose you are sampling from a population with mean µ = 1,065 and standard deviation σ = 500. The sample size is n = 100. What are the expected value and the variance of the sample mean X? Show and explain your work (study guide # 5-41).
#2
Suppose a new estimator for the population mean is discovered. The new

I can not get the true answers, so I need the step by step solutions for these three questions. My problems and necessary information are in the attached file (one question at each page).

1.The central limit theorem says that:
a) Y(bar) is consistent for its mean.
b) Y has a standard normal distribution in large samples.
c) Y(bar) is converge in probability to its mean.
d) the distribution of Y is approximately normally distributed in large samples.
2. If the regression errors are homoskedastic, implies: