Independent random samples X1, X2, . . . , Xn are from
exponential distribution with pdfs 
, xi > 0, where λ is fixed but unknown. Let 
. Here we have a relative large sample size n = 100.
(ii) Notice that the population mean here is µ = E(X1) = 1/λ ,
population variance σ^2 = Var(X1) = 1/λ^2 is unknown. Assume the
sample standard deviation s = 10, sample average 
= 5, construct a 95% large-sample approximate confidence interval
for λ. You may use the following standard normal quantile values:
z0.005 = 2.58, z0.01 = 2.33, z0.025 = 1.96, z0.05 = 1.64, z0.1 =
1.28.
Homework Answers
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Independent random samples X1, X2, . . . , Xn are from
exponential distribution with pdfs...
Let X1, . . . , Xn be a random sample from a triangular probability distribution...
Let X1, . . . , Xn be a random sample from
a triangular probability distribution whose density function and
moments are:
fX(x) =
* I{0
x
b}
a. Find the mean µ of this probability
distribution.
b. Find the Method Of Moments estimator µ(hat) of µ.
c. Is µ(hat) unbiased?
d. Find the Median of this probability distribution.
I will thumbs up any portion or details of how to do this
problem, thanks!
We were unable to transcribe this...
Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State...
Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State whether each of the following statements are true or false, fully justifying your answer. (a) T =(n/n-1)X is a consistent estimator of µ. (b) T = is a consistent estimator of µ (assuming n7). (c) T = is an unbiased estimator of µ. (d) T = X1X2 is an unbiased estimator of µ^2. We were unable to transcribe this imageWe were unable to transcribe...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) =...
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) = for x>0 (a) Use method of moments to find estimators for µ and µ^2 . (b) What is the log likelihood as a function of µ after observing X1 = x1, . . . , Xn = xn? (c) Find the MLEs for µ and µ^2 . Are they the same as those you find in part (a)? (d) According to the Central Limit...
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
Let X1, X2, ..., Xn be a random sample of size n from the distribution with...
Let X1, X2, ..., Xn be a random sample of size n from the
distribution with probability density function
To answer this question, enter you answer as a formula. In
addition to the usual guidelines, two more instructions for this
problem only : write
as single variable p and
as m. and these can be used as inputs of functions as usual
variables e.g log(p), m^2, exp(m) etc. Remember p represents the
product of
s only, but will not work...
Let X1,X2,...,Xn denote independent and identically distributed random variables with variance 2. Which of the following...
Let X1,X2,...,Xn denote independent and identically distributed random variables with variance 2. Which of the following is sucient to conclude that the estimator T = f(X1,...,Xn) of a parameter ✓ is consistent (fully justify your answer): (a) Var(T)= (b) E(T)= and Var(T)= . (c) E(T)=. (d) E(T)= and Var(T)= We were unable to transcribe this imageWe were unable to transcribe this imageoe We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
Let X1, X2, ..., Xn be a random sample from X which has pdf depending on...
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Let be a sample (size n=1) from the exponential distribution, which has the pdf , where...
Let be a sample (size n=1) from the exponential distribution, which has the pdf , where is an unknown parameter. Let's define a statistic as . Is a sufficient statistic for ? We were unable to transcribe this imagef(x: λ) = Xe We were unable to transcribe this imageT(X) = 1122 T(X) We were unable to transcribe this image
Suppose is a random sample from exponential distribution having unknown mean . We wish to test vs. . Consider the following tests: Test 1: Reject if and only if ; Test 2: Reject if and only if...
Suppose is a random sample from exponential distribution having unknown mean . We wish to test vs. . Consider the following tests: Test 1: Reject if and only if ; Test 2: Reject if and only if Find the power of each test at . We were unable to transcribe this imageWe were unable to transcribe this imageHo : θ = 4 We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this...
Let be a sample (size n = 1) from the exponential distribution, which has the pdf...
Let be a sample (size n = 1) from the exponential distribution, which has the pdf where is an unknown parameter. Let's define a statistic as . Is a sufficient statistic for ? We were unable to transcribe this imagef(x: λ) = Xe We were unable to transcribe this imageT(X) = 11>2 T(X) We were unable to transcribe this image
Let X1, . . . , Xn be a random sample from
a triangular probability distribution whose density function and
moments are:
fX(x) =
* I{0
x
b}
a. Find the mean µ of this probability
distribution.
b. Find the Method Of Moments estimator µ(hat) of µ.
c. Is µ(hat) unbiased?
d. Find the Median of this probability distribution.
I will thumbs up any portion or details of how to do this
problem, thanks!
We were unable to transcribe this...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
Let X1, X2, ..., Xn be a random sample of size n from the
distribution with probability density function
To answer this question, enter you answer as a formula. In
addition to the usual guidelines, two more instructions for this
problem only : write
as single variable p and
as m. and these can be used as inputs of functions as usual
variables e.g log(p), m^2, exp(m) etc. Remember p represents the
product of
s only, but will not work...
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
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