Question

Choose the all correct things

Sn=X1+X2+...+Xn

1. E[Sn]=2n, when RV X1,...,Xn is iid and E[X]=2

2. VAR[Sn]=3n, when RV X1,...,Xn is iid and VAR[X]=3

3. VAR[Sn]=2n is always correct, when RV X1,...,Xn is identically distributed and VAR[X]=2

4 .Sn is an Exponential distribution with a mean of 2n and a variance of 3n ,when RV X1,...,Xn is iid and E[X]=2, VAR[X]=3 Exponential distribution

Answer #1

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

Let X1, X2, . . . , Xn be iid exponential random variables with
unknown mean β.
(1) Find the maximum likelihood estimator of β.
(2) Determine whether the maximum likelihood estimator is
unbiased for β.
(3) Find the mean squared error of the maximum likelihood
estimator of β.
(4) Find the Cramer-Rao lower bound for the variances of
unbiased estimators of β.
(5) What is the UMVUE (uniformly minimum variance unbiased
estimator) of β? What is your reason?
(6)...

X1 and X2 are iid exponential (2) random
variables and Z=max(X1 , X2). What is
E[Z]?
(Hint: Find CDF and then PDF of Z)
A. 3/2
B. 3
C. 1/2
D. 3/4

let X1 X2 ...Xn-1 Xn be independent exponentially distributed
variables with mean beta
a). find sampling distribution of the first order statistic
b). Is this an exponential distribution if yes why
c). If n=5 and beta=2 then find P(Y1<=3.6)
d). find the probability distribution of Y1=max(X1, X2, ...,
Xn)

Problem 1 Let X1, · · · , Xn IID∼ p(x; θ) = 1/2 (1 +θx), −1 <
x < 1, −1 < θ < 1. 1. Estimate θ using the method of
moments. 2. Show that the above MoM is consistent by showing it’s
mean square error converges to 0 as n goes to infinity. 3. Find its
asymptotic distribution.

A random sample X1, X2, . . . , Xn is drawn from a population
with pdf. f(x; β) = (3x^2)/(β^3) , 0 ≤ x ≤ β 0, otherwise
(a) [6] Find the pdf of Yn, the nth order statistic of the
sample.
(b) [4] Find E[Yn].
(c) [4] Find Var[Yn].
(d)[3] Find the mean squared error of Yn when Yn is used as a
point estimator for β
(e) [2] Find an unbiased estimator for β.

Included all steps. Thanks
The random variable X is uniformly distributed in the interval
[0, α] for some α > 0.
Parameter α is fixed but unknown. In order to estimate α, a
random sample X1, X2, . . . , Xn of independent and identically
distributed random variables with the same distribution as X is
collected, and the maximum value Y = max{X1, X2, ..., Xn} is
considered as an estimator of α.
(a) Derive the cumulative distribution function...

5. Consider a simple case with only four independently and
identically distributed (iid) observations, X1, X2, X3, X4, on a
random variable X. Consider these two estimators:
µˆ1 = 1/12 (2X1 + 4X2 + 4X3 + 2X4), µˆ2 = 1/12 (X1 + 5X2 + 5X3 +
X4).
a Show that each is unbiased, and that one is more efficient
than the other.
b Show that the usual sample mean is more efficient than either.
Explain why the others given above...

Let X1,…, Xn be a sample of iid
Exp(?1, ?2) random variables with common pdf
f (x; ?1, ?2) =
(1/?1)e−(x−?2)/?1 for x
> ?2 and Θ = ℝ × ℝ+.
a) Show that S = (X(1), ∑ni=1
Xi ) is jointly sufficient for (?1, ?2).
b) Determine the pdf of X(1).
c) Determine E[X(1)].
d) Determine E[X2(1) ].
e ) Is X(1) an MSE-consistent estimator of
?2?
f) Given S = (X(1), ∑ni=1
Xi )is a complete sufficient statistic...

The random variables X1 and X2 both follow
normal distributions. The mean of X1 is
E(X1)=5, and its variance is V(X1)=2 The mean
of X2 is E(X2)=9, and its variance is
V(X2)=3. If Y is a random variable such that Y =
3X1+5X2, what is P(Y<70)?
A student takes 4 measurements and finds that the mean is 64 and
the sample variance is 81. What is the sample standard
deviation
For a random variable X, which statement is most likely...

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