Question

Prove: Let x and y be bounded sequences such that x_{n}
≤ y_{n} for all n ∈ N. Then lim sup_{n}→∞
x_{n} ≤ lim sup_{n}→∞ y_{n} and lim
inf_{n}→∞ x_{n} ≤ lim inf_{n}→∞
y_{n}.

Answer #1

Let
( xn) and (yn) be sequence with xn converge to x and yn converge to
y. prove that for dn=((xn-x)^2+(yn-y)^2)^(1/2), dn converge to
0.

Let
x = {x} and y ={y} represent bounded sequences of real numbers, z =
x + y, prove the following: supX + supY = supZ where sup represents
the supremum of each sequence.

If lim Xn as n->infinity = L and lim Yn as n->infinity =
M, and L<M then there exists N in naturals such that Xn<Yn
for all n>=N

Let {xn} be a non-decreasing sequence and assume that xn goes to
x as n goes to infinity. Show that for all, n in N (naturals), xn
< x. Formulate and prove an analogous result for a
non-increasing sequences.

Let S be the collection of all sequences of real numbers and
define a relation on S by {xn} ∼ {yn} if and only if {xn − yn}
converges to 0.
a) Prove that ∼ is an equivalence relation on S.
b) What happens if ∼ is defined by {xn} ∼ {yn} if and only if
{xn + yn} converges to 0?

Let 0 < θ < 1 and let (xn) be a sequence where
|xn+1 − xn| ≤ θn for n
= 1, 2, . . ..
a) Show that for any 1 ≤ n < m one has |xm −
xn| ≤ (θn/ 1-θ )*(1 − θ m−n ).
Conclude that (xn) is Cauchy
b)If lim xn = x* , prove the following error in
approximation (the "error in approximation" is the same as error
estimation in Taylor Theorem) in t:...

Let (X1, Y1), . . . ,(Xn, Yn), be a random sample from a
bivariate normal distribution with parameters µ1, µ2, σ2 1 , σ2 2 ,
ρ. (Note: (X1, Y1), . . . ,(Xn, Yn) are independent). What is the
joint distribution of (X ¯ , Y¯ )?

If Xn is a cauchy sequence and Yn is also a cauchy sequence,
then prove that Xn+Yn is also a cauchy sequence

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

Let X1, X2, ..., Xn be a random sample (of size n) from U(0,θ).
Let Yn be the maximum of X1, X2, ..., Xn.
(a) Give the pdf of Yn.
(b) Find the mean of Yn.
(c) One estimator of θ that has been proposed is Yn. You may
note from your answer to part (b) that Yn is a biased estimator of
θ. However, cYn is unbiased for some constant c. Determine c.
(d) Find the variance of cYn,...

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