First order moments , estimation , estimators
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Probability and Statistics
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....
(See attached file for full problem description)
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The expert calculated the first order moments, estimators and estimation.
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Please see the attached files.
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)
∂Log L = 0
∂ θ
∑ 1/ (θ+1) + ∑ log( xi) = 0
n/ (θ+1) = - ∑ log( xi)
n/ ∑ log( xi) = -(θ+1)
θ = -1 - n / log (Π ...
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