Question
In: Statistics and Probability
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?
Solutions
Expert Solution
the first part of the solution
for (a).
the second part of the solution
showing the exact distribution and approximate distribution of
Y.
the solution for (b). Showing
that Y is the MVUE of ∅.
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. (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).
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)....
2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 − x), 0 <...
2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 −
x), 0 < x < 1,
0 elsewhere.
(a) Find the value of c such that fX(x) is indeed a PDF.
(b) Find P(−0.5 < X < 0.3).
(c) Find the median of X.
A random variable Y is a function of random variable X, where y=x^2 and fx(x)=(x+1)/2 from...
A random variable Y is a function of random variable X, where
y=x^2 and fx(x)=(x+1)/2 from -1 to 1 and =0 elsewhere. Determine
fy(y). In this problem, there are two x values for every y value,
which means x=T^-1(y)= +y^0.5 and -y^0.5. Be sure you account for
both of these. Ans: fy(y)=0.5y^-0.5
Let X be a random variable. Suppose X ∼ Exp(0.5). The area under the pdf of...
Let X be a random variable. Suppose X ∼ Exp(0.5). The area under
the pdf of the Exp(0.5) distribution above the interval [0, 2] is
0.6321. What is the value of the cumulative distribution function
FX of X at x = 2?
The (mixed) random variable X has probability density function (pdf) fX (x) given by: fx(x)=0.5δ(x−3)+ {...
The (mixed) random variable X has probability density function
(pdf) fX (x) given by:
fx(x)=0.5δ(x−3)+ { c.(4-x2), 0≤x≤2
0, otherwise
where c is a constant.
(a) Sketch fX (x) and find the constant c.
(b) Find P (X > 1).
(c) Suppose that somebody tells you {X > 1} occurred. Find
the conditional pdf fX|{X>1}(x), the pdf of X given
that {X > 1}.
(d) Find FX(x), the cumulative distribution function of X.
(e) Let Y = X2 . Find...
Let X have the pdf fX(x) = 3(1 − x) 2 , 0 < x <...
Let X have the pdf fX(x) = 3(1 − x) 2 , 0 < x < 1.
(a) Find the pdf of Y = (1 − X) 3 . Specify the distribution of
Y (name and parameter values). (b) Find E(Y ) and Var(Y ).
using R :- A Gumbel random variable X has distribution function FX (x) = exp (−e^−x)....
using R :-
A Gumbel random variable X has distribution function
FX (x) = exp (−e^−x).
a) Give a graph of FX and explain using this plot why FX is a
valid cumulative probability distri-
bution function.
(b) Find the values of the first and third quartiles and median
X and show their values on the graph.
(c) Make a table of x and FX (x) for x equal to the integers
from −2 to 5.
(d) Find the probabilities...
Let X be a continuous random variable with pdf: f(x) = ax^2 − 2ax, 0 ≤...
Let X be a continuous random variable with pdf: f(x) = ax^2 −
2ax, 0 ≤ x ≤ 2
(a) What should a be in order for this to be a legitimate
p.d.f?
(b) What is the distribution function (c.d.f.) for X?
(c) What is Pr(0 ≤ X < 1)? Pr(X > 0.5)? Pr(X > 3)?
(d) What is the 90th percentile value of this distribution?
(Note: If you do this problem correctly, you will end up with a
cubic...
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