Let Y1, Y2, . . ., Yn be a
random sample from a uniform distribution on the interval (θ - 2,
θ).
a) Show that Ȳ is a biased estimator of θ. Calculate the
bias.
b) Calculate MSE( Ȳ).
c) Find an unbiased estimator of θ.
d) What is the mean square error of your unbiased estimator?
e) Is your unbiased estimator a consistent estimator of θ?
Let Y = (Y1, Y2,..., Yn) and
let θ > 0. Let Y|θ ∼ Pois(θ). Derive the posterior density of θ
given Y assuming the prior distribution of θ is Gamma(a,b) where a
> 1. Then find the prior and posterior means and prior and
posterior modes of θ.
Suppose that Y1 ,Y2 ,...,Yn is
a random sample from distribution Uniform[0,2].
Let Y(n) and Y(1) be the order
statistics.
(a) Find E(Y(1))
(b) Find the density of (Y(n) − 1)2
(c) Find the density of Y(n) − Y (1)
Let
Y1, Y2, ..., Yn be a random sample from an exponential distribution
with mean theta. We would like to test H0: theta = 3 against Ha:
theta = 5 based on this random sample.
(a) Find the form of the most powerful rejection region.
(b) Suppose n = 12. Find the MP rejection region of level
0.1.
(c) Is the rejection region in (b) the uniformly most powerful
rejection region of level 0.1 for testing H0: theta = 3...
. Let Xl' ... ' Xn rv Uniform(0,θ) and let θˆ = max{Xl , ... ,
Xn}. Find the bias, se, and MSE of this estimator. + If we assume
that θˆ is asymptotically normal, what is the 90% percentile
confidence interval? Suppose my data was 1.0 2.0 3.0 4.0 5.0 and
report some numbers.
Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show
that for any α ∈ (0, 1), there is a cn,α, such that
[max(X1,...,Xn),cn,α max(X1,...,Xn)] is a 1−α confidence interval
of θ.
1. . Let X1, . . . , Xn, Y1, . . . , Yn be mutually independent
random variables, and Z = 1 n Pn i=1 XiYi . Suppose for each i ∈
{1, . . . , n}, Xi ∼ Bernoulli(p), Yi ∼ Binomial(n, p). What is
Var[Z]?
2. There is a fair coin and a biased coin that flips heads with
probability 1/4. You randomly pick one of the coins and flip it
until you get a...