I have four problems in the attached file. I would like the solutions to be very detailed as much as possible because this homework will be my review for the midterm. My own solutions are wrong so need someone to help me solve this....

xi = number of teachers absent on day i, for i=1,2,...,50. (this is wrong)
It is xi = number of days out of 50 days i teachers are absent .

The calculations are in excel file.

22a)
The first order moment about origin is given by

1 1
∫ x*(θ+1) * x ˆ θ dx = [ (θ+1) x ˆ (θ+2) / (θ+2) ] = θ+1 / θ+2
0 0
the first order moment is nothing but mean .
_
x = θ+1 / θ+2
_
(θ+2) x = θ+1
_ _
θ = 2 x -1 / 1 - x

Finding the mean of the given data
.92+.79+.9+.65+.86+.47+.73+.97+.94+.77 / 10 = 8/10 = .8
Subsituting we get θ = 1.6 - 1 / 1 - .8 = .6/.2 = 3

b)Maximum likelihood estimator
n
L = Π f(xi , θ)
.i=1

n
Π (θ+1) * xi ˆ θ
.i=1

taking logs on both sides we get

Log L = ∑(log((θ+1) * xi ˆ θ ))
∑log (θ+1) + ∑ θ log( xi)

Please see the attached file for full problem.
a) Using the results for the gamma distribution, show that s^2 is an unbiased estimator of sigma^2. Also state its variance.
b) Consider a statistic of the form T_k = ks^2, for some constant k. Find expressions for the bias and variance of T_k as an estimator of sigma^2. Hene,

From a Poisson (θ) distribution a random sample X1, X2, ... , Xn is selected. Given that that S = X1 + ... + Xn is sufficient for θ and also has the Poisson(nθ) distribution, we can define gr,k(s) by
gr,k(s) = {s!/(s - r)!} n-r {1 - (k/n)}s-r, s = r, r + 1, ... , 0 otherwise,
in which r = 0, 1, 2, ...

Let Y1order statistics of a random sample of size n from the uniform distribution of the continuous type over the closed interval [theta- rho, theta+ rho]. Find the maximum likelihood estimators for theta and rho. Are these two unbiased estimators?

Please see the attachment for probability questions.
Find the means and variances of these estimators.
Hence answer the following questions, giving reasons for your answers.
a) Which of these estimators are unbiased
b) Which of the unbiased estimators is the most efficient as an estimator for μ?
c) Which of these es

Random variables x and y have the joint density function
fxy(x,y)={ (a(2x+y)^2)/20 -1 < x < 1 -3 < y < 3}
{ 0 elsewhere }
a)find the value of the constant a
b)find all the second ordermoments of X and Y
c)what are variances of X and Y
d)what is the correlation coefficient

9. (Method of Moments and Maximum Likelihood, 8 points)
The exponential random variable is given by
f(x) = lamba(e^-lamba(x)), x (greater or equal to) 0.
a) Recall that EX = 1/lamba. Use the method of moments to estimate lamba.
b) Find the maximum likelihood estimate of lamba.

22. Let X denote the proportion of alloted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of X is ...
a. Use method of moments to obtain an estimator ...
b. Obtain the maximum likelihood esitmator ...
Please see attachment for complete question. Thank you!

I am having trouble with some estimators. I have defined a random variable Z such that Z=Y/X, where X and Y are random variables. I know that E(Y given X)=theta*X, where theta is an unknown parameter. I have worked out that E(Z)=theta.
Now I am having trouble with an estimator defined as W=Ybar/Xbar, where Ybar and Xbar refer

Suppose that X is a discrete random variable with P(X=1) = θ and P(X=2)= 1- θ. Three independent observations of X are made: X1 = 1, X2 = 2, X3 = 2.
Find the Method of Moments estimate of θ. Please show your work.