Suppose U is uniform on (0,1). Let Y = U(1 − U). (a) Find P(Y
> y) for 0 < y < 1/4. (b) differentiate to get the density
function of Y . (c) Find an increasing function g(u) so that g(U )
has the same distribution as U (1 − U ).
Let X1 and X2 be uniformly distributed in the region
[−1,1]×[0,1]∪(1,2]×(1,3].
1. Find Joint and Marginal pdf of X1 and X2.
2.Find V (X1 + 3X2) ,( I‘ve asked this question already, but the
person didn't give me the correct answer, so please, dont waste my
question, if you dont know how to answer this question.)
Chapter P, Section 5, Exercise 134
Find the specified areas for a N(0,1) density.
(a) The area below z=0.9
Round your answer to four decimal places.
(b) The area above z=1.3
Round your answer to four decimal places.
(c) The area between z=1.78 and z=1.21
Round your answer to four decimal places.
6. Let X1, X2, ..., X101 be 101 independent U[0,1] random
variables (meaning uniformly distributed on the unit interval). Let
M be the middle value among the 101 numbers, with 50 values less
than M and 50 values greater than M.
(a). Find the approximate value of P( M < 0.45 ).
(b). Find the approximate value of P( | M- 0.5 | < 0.001 ),
the probability that M is within 0.001 of 1/2.
Let X1, X2 ,X3 ∼U[0,1] be
independent, and letX(1),X(2),X(3) be the
order statistics, i.e., X(1) < X(2) < X(3)
(a) Find f(1)(x), f(2)(y) and
f(3)(z).
(b) Verify that the joint density of X(1) and
X(2) is
f(1,2)(x,y)=6(1−y), 0
(c) Find the conditional density of X(1) given X(2).
(d) Find E(X(1)+X(3 )/
X(2))
Let (Un, U, n>1) be asequence of random variables such that
Un and U are independent, Un is N(0, 1+1/n), and U is N(0,1), for
each n≥1.
Calculate p(n)=P(|Un-U|<e), for all e>0.
Please give details as much as possible
Let Y 1 ,...,Y n be a sample from the
density f(y) = λ 2 ye −λy , y > 0 where λ
> 0 is an
unknown parameter.
(a) Find an estimator 'λ 1 of λ by Method of
Moments
(b) Find an estimator 'λ 2 of λ by Method of Maximum
Likelihood.
(c) Find an estimator 'λ 3 of λ that is a Sufficient
estimator. Can you construct a
Minimal Variance Unbiased Estimator? Justify.
Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2)
− (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4).
b) Make a conjecture that gives a closed form (i.e., not a
summation) formula for the value of S(n). c) Use induction to prove
your conjecture is correct.