Question
In: Statistics and Probability
For a random variable X with a Cauchy distribution with θ = 0 , so that...
For a random variable X with a Cauchy distribution with θ = 0 ,
so that
f(x) =(1/ π)/( 1 + x^2) for -∞ < x < ∞
(a) Show that the expected value of the random variable X does not
exist.
(b) Show that the variance of the random variable X does not
exist.
(c) Show that a Cauchy random variable does not have finite moments
of order greater than or equal to one.
Solutions
orchestra answered 1 year agoRelated Solutions
The random variable X is distributed with pdf fX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0....
The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0.
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Crameer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).
The random variable X is distributed with pdf fX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0....
The random variable X is distributed with pdf fX(x,
θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
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The random variable X is distributed with pdf fX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0....
The random variable X is distributed with pdf
fX(x, θ) = c*x*exp(-(x/θ)2), where x>0
and θ>0. (Please note the equation includes the term
-(x/θ)2 or -(x/θ)^2 if you cannot read that)
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB).
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= c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please
note the equation includes the term -(x/θ)2 - that is
-(x/θ)^2 if your computer doesn't work)
a) What is the constant c?
b) We consider parameter θ is a number. What is MLE and MOM of
θ? Assume you have an i.i.d. sample. Is MOM unbiased?
c) Please calculate the Cramer-Rao Lower Bound (CRLB). Compare
the variance of MOM with Crameer-Rao Lower Bound (CRLB)....
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X 0 1 2 3
P(X) 0.25 0.40 0.20 0.15
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a.
0.9875
b.
3.0000
c.
0.5623
d.
0.9937
e.
0.6000
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1
2
P(X)
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2/8
3/8
2/8
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