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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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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.

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
L = Π f(xi , θ)

Π (θ+1) * xi ˆ θ

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 ...

Solution Summary

The expert calculated the first order moments, estimators and estimation.

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Direction of a Moving Particle in a Charged Particle

1) The neutron is a particle with zero charge. However, it has a nonzero magnetic moment of 9.66 × 10−27Am2. A possible explanation for this is the circular motion of 'quarks' - fundamental subatomic particles. The neutron is believed to consist of an "up" quark with a charge of +2e/3 and two "down" quarks each of charge −e/3 (note net charge is still zero). If the quarks are in motion they can create a magnetic moment. In particular, the magnetic moment produced by a circular current is µ = IA, where I is the current and A is the area of the circle. The radius of the neutron is 1.2 × 10−15m and so this seems a reasonable estimate for the radius of the orbits. If the up quark moves around in one direction, and the down quarks move in the opposite direction, all with the same speed v how fast will they have to be moving to generate the observed magnetic moment?

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