: Let X1, X2, . . . , Xn be a random sample from the normal
distribution N(µ, 25). To test the hypothesis H0 : µ = 40 against
H1 : µne40, let us define the three critical regions: C1 = {x¯ : ¯x
≥ c1}, C2 = {x¯ : ¯x ≤ c2}, and C3 = {x¯ : |x¯ − 40| ≥ c3}. (a) If
n = 12, find the values of c1, c2, c3 such that the size of...
. Let X1, X2, . . . , Xn be a random sample from a normal
population with mean zero but unknown variance σ 2 . (a) Find a
minimum-variance unbiased estimator (MVUE) of σ 2 . Explain why
this is a MVUE. (b) Find the distribution and the variance of the
MVUE of σ 2 and prove the consistency of this estimator. (c) Give a
formula of a 100(1 − α)% confidence interval for σ 2 constructed
using the...
Let X1,X2, . . . , Xn be a random sample from the uniform
distribution with pdf f(x; θ1, θ2) =
1/(2θ2), θ1 − θ2 < x <
θ1 + θ2, where −∞ < θ1 < ∞
and θ2 > 0, and the pdf is equal to zero
elsewhere.
(a) Show that Y1 = min(Xi) and Yn = max(Xi), the joint
sufficient statistics for θ1 and θ2, are
complete.
(b) Find the MVUEs of θ1 and θ2.
Let X1, X2, . . . , Xn be a random sample of size n from a
Poisson distribution with unknown mean µ. It is desired to test the
following hypotheses
H0 : µ = µ0
versus H1 : µ not equal to µ0
where µ0 > 0 is a given constant. Derive the likelihood ratio
test statistic
We have a random sample of observations on X: x1, x2, x3,
x4,…,xn. Consider the following estimator of the population mean:
x* = x1/2 + x2/4 + x3/4. This estimator uses only the first three
observations.
a) Prove that x* is an unbiased estimator.
b) Derive the variance of x*
c) Is x* an efficient estimator? A consistent estimator?
Explain.
Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf
f(x; θ) = (x/θ) * e^x^2/(2θ) for x > 0
(a) It can be shown that E(X2 P ) = 2θ. Use this fact to construct an unbiased estimator of θ based on n i=1 X2 i .
(b) Estimate θ from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions:
16.88 10.23 4.59 6.66...
Let X1, X2, ..., Xn be a random sample of size from a
distribution with probability density function
f(x) = λxλ−1 , 0 < x < 1, λ > 0
a) Get the method of moments estimator of λ. Calculate the
estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3.
b) Get the maximum likelihood estimator of λ. Calculate the
estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3.
. Let X1, X2, ..., Xn be a random sample of size 75 from a
distribution whose probability distribution function is given by
f(x; θ) = ( 1 for 0 < x < 1, 0 otherwise. Use the central
limit theorem to approximate P(0.45 < X < 0.55)