Question
In: Statistics and Probability
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. b. Repeat part a) but assume that Ω = {−1, 0, 1}
Solutions
orchestra answered 1 year agoRelated Solutions
LetXbe a discrete random variable with the following PMF: P(X=x) = 0.3 for x = 1,...
LetXbe a discrete random variable with the following PMF:
P(X=x) =
0.3 for x = 1,
a for x = 2,
0.5 for x= 3,
0 otherwise
(a) Find the value of a.
(b) Find Fx(x), the CDF of X.
(c) What is the value of Fx(2)?
(d) Find E[X].
(e) FindE[X^2].
(f) Find V ar(X)
----------------------
A certain area of the eastern United States is, on average,hit
by 2 hurricanes a year. Find the probability that the area will...
A discrete random variable named X has the following pmf (probability mass function): X P(x) 1...
A discrete random variable named X has the following
pmf (probability mass function):
X
P(x)
1
0.6
3
0.2
7
0.1
11
0.1
17a) Find P(X>6)
17b) What is the probability two independent observations of X
will both equal 1?
17c) Find the population mean also known as E(X)= the expected
value of
17d) Find the population variance of X with both the “regular”
formula and “convenient”
formula
Make a table and show your work. You should get the same...
Let the random variable X and Y have the joint pmf f(x, y) = , c...
Let the random variable X and Y have the joint pmf f(x, y) = , c
xy 2 where x = 1, 2, 3; y = 1, 2, x + y ≤ 4 , that is, (x, y) are
{(1, 1),(1, 2),(2, 1),(2, 2),(3, 1)} .
(a) Find c > 0 .
(b) Find μ . X
(c) Find μ . Y
(d) Find σ . 2 X
(e) Find σ . 2 Y
(f) Find Cov (X, Y )...
Let the random variable and have the joint pmf X Y f(x,y) = {x(y)^2}/c where x...
Let the random variable and have the joint pmf X Y f(x,y) =
{x(y)^2}/c
where x = 1, 2, 3 ; y = 1, 2, x+y<= 4, that is (x,y) are {(1,1),
(1,2), (2,1), (2,2), 3,1)}
(a) Find . c > 0
(b) Find . μX
(c) Find . μY
(d) Find . σ2 X
(e) Find . σ2 Y
(f) Find Cov . (X,Y )
(g) Find p , Corr . (x,y)
(h) Are and X and Y independent
is 1. f(x/θ) = 2x/θ^2 complete sufficient statistic 2. f(x/θ) =((logθ) θ^x ) / (θ -1) complete...
is 1. f(x/θ) = 2x/θ^2 complete sufficient statistic
2. f(x/θ) =((logθ) θ^x ) / (θ -1) complete sufficient
statistic?
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).
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)....
Suppose X is a discrete random variable with mean μ and variance σ^2. Let Y =...
Suppose X is a discrete random variable with mean μ and variance
σ^2. Let
Y = X + 1.
(a) Derive E(Y ).
(b) Derive V ar(Y ).
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