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 2 years 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)....
A random variable (RV), x is distributed with a pdf as follows: fX(x) = [C ×...
A random variable (RV), x is distributed with a pdf as
follows:
fX(x) = [C × x × (1 – x)]
0 ≤ x ≤ a
(C and
a are constants)
=
0
otherwise
Hence, determine the following: (i) C; (ii) cumulative
probability of x in the range, [(u × a) ≤ x ≤ (v × a)]; where u and
v are fractions; (iii) E[x]; (iv) second moment of x ; and, (v)
standard deviation of x. Assume:...
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...
1) The PDF of a Gaussian random variable is given by fx(x). fx(x)= (1/(3*sqrt(2pi) )*e^((x-4)^2)/18 determine...
1) The PDF of a Gaussian random variable is given by fx(x).
fx(x)= (1/(3*sqrt(2pi)
)*e^((x-4)^2)/18
determine
a.) P(X > 4) b). P(X > 0). c). P(X
< -2).
2) The joint PDF of random variables
X and Y is given by
fxy(x,y)=Ke^-(x+y), x>0 , y>0
Determine
a. The constant k.
b. The marginal PDF fX(x).
c. The marginal PDF fY(y).
d. The conditional PDF fX|Y(x|y). Note
fX|Y(x|y) =
fxy(x,y)/fY(y)
e. Are X
and Y independent.
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 ).
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