Let Y1 < Y2 < Y3 <
Y4 be the order statistics of a random sample of size n
= 4 from a distribution with pdf f(x) =
3X2, 0 < x < 1, zero
elsewhere.
(a) Find the joint pdf of Y3 and
Y4.
(b) Find the conditional pdf of Y3,
given Y4 = y4.
(c) Evaluate E(Y3|y4)
Let Y1 < Y2 < Y3 <
Y4 < Y5 denote the order statistics of a
random sample of size 5 from a distribution having pdf f(x) = e−x,
0 < x < ∞, zero elsewhere.
show that Y4 and Y5 – Y4 are
independent.
Hint: First find the joint pdf of Y4 and
Y5.
Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1
0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1,
Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).
Let Y1, Y2, Y3, and
Y4be independent, identically distributed random
variables from a population with a mean μ and a variance
σ2. Consider a different estimator of μ:
W
= Y1+ Y2+
Y3+ Y4.
Let Y1, Y2, Y3, and
Y4be independent, identically distributed random
variables from a population with a mean μ and a variance
σ2. Consider a different estimator of μ:
W = 1/8 Y1+ 1/3
Y2+ 1/6 Y3+ 3/8 Y4.
This is an example of a weighted
average of the Yi.
Show...
If you conduct and experiment 1500 times independently,
i=1,2,3,...1500. Let y1, y2.... yN be i.i.d observations
from this experiment, yi=1 if heads with a probability of β; yi=0
if tails with a probability of 1-β. If you get 600 heads and 900
tails, what is the βMLE?
A.
0.4
B.
0.5
C.
0.6
D.
0.7
Let c(y1, y2) = y1 + y2 + (y1y2)^ −(1/3). Does this cost
function have economies of scale for y1? What about economies of
scope for any strictly positive y1 and y2. Hint, economies of scope
exist if for a positive set of y1 and y2, c(y1, y2) < c(y1, 0) +
c(0, y2). [Hint: Be very careful to handle the case of y2 = 0
separately.]
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 F= (x2 +
y + 2 + z2) i + (exp(
x2 ) + y2)
j + (3 + x) k . Let a
> 0 and let S be part of the spherical
surface x2 + y2 +
z2 = 2az + 15a2
that is above the x-y plane and the disk formed in the
x-y plane by the circular intersection between the sphere
and the plane. Find the flux of F outward across
S.