Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn)
where Y...
Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn)
where Y | X = x is modeled by Y=β0+β1x+ε, ε∼N(0,σ^2), where
β0,β1and σ^2 are unknown. Let β1 denote the mle of β1. Derive
V(βhat1).
Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent
random samples, with the variables Xi normally distributed with
mean μ1 and variance σ12 and the variables Yi normally distributed
with mean μ2 and variance σ22. The difference between the sample
means, X − Y, is then a linear combination of m + n normally
distributed random variables and, by this theorem, is itself
normally distributed.
(a) Find E(X − Y).
(b) Find V(X − Y).
(c)...
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...
The parametric equations
x = x1 +
(x2 −
x1)t, y
= y1 +
(y2 −
y1)t
where
0 ≤ t ≤ 1
describe the line segment that joins the points
P1(x1,
y1)
and
P2(x2,
y2).
Use a graphing device to draw the triangle with vertices
A(1, 1), B(4, 3), C(1, 6). Find the
parametrization, including endpoints, and sketch to check. (Enter
your answers as a comma-separated list of equations. Let x
and y be in terms of t.)
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?
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 X1,X2,...,Xn and Y1,Y2,...,Ym be independent random samples
drawn from normal distributions with means μX and μY ,
respectively, and with the same 2 known variance σ . Use the
generalized likelihood ratio criterion to derive a test procedure
for choosing between H0:μX =μY and H1:μX not equal to μY.
1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1
−x2|+|y1 −y2|.
(a) Prove that (R2,ρ) is a metric space.
(b) In (R2,ρ), sketch the open ball with center (0,0) and radius
1. 2. Let {xn} be a sequence in a metric space (X,ρ). Prove that if
xn → a and xn → b for some a,b ∈ X, then a = b.
3. (Optional) Let (C[a,b],ρ) be the metric space discussed in
example 10.6 on page 344...
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1
+ X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y =
(Y1,Y2,Y3)′ using : Multivariate normal distribution
properties.
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 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...