Question

Let *Y _{1},
Y_{2,} …, Y_{n}*denote a random sample of size
n from a population whose density is given by

f(y) = 5y^4/theta^5 0<y<theta

0 otherwise

a) Is an unbiased estimator of θ?

b) Find the MSE of Y bar

c) Find a function of that is an unbiased estimator of θ.

Answer #1

Let Y1, Y2, . . ., Yn be a
random sample from a Laplace distribution with density function
f(y|θ) = (1/2θ)e-|y|/θ for -∞ < y < ∞
where θ > 0. The first two moments of the distribution are
E(Y) = 0 and E(Y2) = 2θ2.
a) Find the likelihood function of the sample.
b) What is a sufficient statistic for θ?
c) Find the maximum likelihood estimator of θ.
d) Find the maximum likelihood estimator of the standard
deviation...

Let Y1, Y2, ... Yn be a random sample of an exponential
population with parameter θ. Find the density function of the
minimum of the sample Y(1) = min(Y1, Y2, ..., Yn).

Let Y1,Y2.....,Yn be independent ,uniformly distributed random
variables on the interval[0,θ].，Y(n)=max(Y1,Y2,....,Yn)，which is
considered as an estimator of θ. Explain why Y is a good estimator
for θ when sample size is large.

Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function f(x; θ) = (θ 4/6)x 3 e −θx if 0 < x
< ∞ and 0 otherwise where θ > 0
. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
sufficient statistic for θ. b. Compute E(1/Y ) and find the
function of Y which is the unique minimum variance unbiased
estimator of θ.
b. Compute...

Problem 3. Let Y1, Y2, and Y3 be independent, identically
distributed random variables from a population with mean µ = 12 and
variance σ 2 = 192. Let Y¯ = 1/3 (Y1 + Y2 +
Y3) denote the average of these three random
variables.
A. What is the expected value of Y¯, i.e., E(Y¯ ) =? Is Y¯ an
unbiased estimator of µ?
B. What is the variance of Y¯, i.e, V ar(Y¯ ) =?
C. Consider a different estimator...

1. (a) Y1,Y2,...,Yn form a random sample from a probability
distribution with cumulative distribution function FY (y) and
probability density function fY (y). Let Y(1) = min{Y1,Y2,...,Yn}.
Write the cumulative distribution function for Y(1) in terms of FY
(y) and hence show that the probability density function for Y(1)
is fY(1)(y) = n{1−FY (y)}n−1fY (y). [8 marks]
(b) An engineering system consists of 5 components connected in
series, so, if one components fails, the system fails. The
lifetimes (measured in...

Let X1,X2,...,X50 denote a random sample of size 50 from the
distribution whose probability density function is given by f(x)
=(5e−5x, if x ≥ 0 0, otherwise If Y = X1 + X2 + ... + X50, then
approximate the P(Y ≥ 12.5).

Let X be the mean of a random sample of size n from a N(θ, σ2)
distribution,
−∞ < θ < ∞, σ2 > 0. Assume that σ2 is known. Show that
X
2 − σ2
n is an
unbiased estimator of θ2 and find its efficiency.

Let Y1, Y2, . . . , Yn denote a random sample from a uniform
distribution on the interval (0, θ). (a) (5 points)Find the MOM for
θ. (b) (5 points)Find the MLE for θ.

Let Y1 < Y2 < Y3 <
Y4 be the order statistics of a random sample of size n
= 5 (Y1 < Y2 < Y3 <
Y4 <Y5). from the distribution having pdf
f(x) = e−x, 0 < x < ∞, zero elsewhere. Find P(Y5 ≥ 3).

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