Question
In: Statistics and Probability
The random variable X is distributed with pdffX(x, θ) = c*x*exp(-(x/θ)2), where x>0 and θ>0. (Please...
The random variable X is distributed with pdffX(x, θ) = 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).
Note: If you do the equation using the term -(2x/θ) it is wrong. If you can't read the equation, please don't do the work. Thanks.
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?
b) Find the best estimator, i.e. MVUE, of θ?
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).
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.
For f(x; θ) = θ exp(-xθ) , x>0 1a) Determine the most powerful critical region for...
For f(x; θ) = θ exp(-xθ) , x>0
1a) Determine the most powerful critical region for testing
H0 θ=θ0 against H1 θ=θ1
(θ1 > θ0) using a random sample of size
n.
1b) Find the uniformly most powerful H0
θ<θ0 against H1 θ>θ1
Suppose X and Y are independent random variables with Exp(θ = 2) distribution. Note that, we...
Suppose X and Y are independent random variables with Exp(θ = 2)
distribution. Note that, we say X ∼ Exp(θ) if its pdf is f(x) = 1/θ
e^(−x/θ) , for x > 0 and θ > 0.
(a) What is the joint probability density function (pdf) of (X,
Y )?
(b) Use the change of variable technique (transformation
technique) to evaluate the joint pdf fW,Z (w, z) of (W, Z), where W
= X −Y and Z = Y ....
a. Suppose that X is a discrete random variable with pmf f(x) = (2 + θ(2...
a. Suppose that X is a discrete random variable with pmf f(x) =
(2 + θ(2 − x))/ 6 , x = 1, 2, 3, where the parameter θ belongs to
the parameter space Ω = (θ : −2 < θ < 2). Suppose further
that a random sample X1, X2, X3, X4 is taken from this
distribution, and the four observed values are {x1, x2, x3, x4} =
{3, 2, 3, 1}. Find the maximum likelihood estimate of θ....
Assume a random variable X with four possible outcomes {1,2,3,4}, each with the probability θ/2, θ/2,...
Assume a random variable X with four possible outcomes
{1,2,3,4}, each with the probability θ/2, θ/2, (1-θ)/3, and
(2-2θ)/3, respectively. We observe the following samples
{1,1,1,2,2,3,3,4,4,4}. Derive the maximum likelihood estimate of
θ.
Does a continuous random variable X exist with E(X − a) = 0, where a is...
Does a continuous random variable X exist with E(X − a) = 0,
where a is the
day of your birthdate? If yes, give an example for its probability
density function.
If no, give an explanation.
(b) Does a continuous random variable Y exist with E
(Y − b)
2
= 0, where b is
the month of your birthdate? If yes, give an example for its
probability density
function. If no, give an explanation.
If the random variable x~N(0,c^2), and g(x)=x^2, find and sketch the distribution and density function of...
If the random variable x~N(0,c^2), and g(x)=x^2, find and sketch
the distribution and density function of the random variable
y=g(x).
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