Let a sequence {xn} from n=1 to infinity satisfy
x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ......
1. Prove that a<=xn<=b for all n>=1
2. Show |x_(n+1) - xn| <= sqrt(b)/(sqrt(a)+sqrt(b)) * |xn -
x_(n-1)| for n=2,3,.....
3. Prove {xn} is a cauchy sequence and hence is convergent
Please show full working for 1,2 and 3.
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...
Let (xn) be a sequence with positive terms. (a) Prove
the following: lim inf xn+1/ xn ≤ lim inf
n√ xn ≤ lim sup n√xn ≤
lim sup xn+1/ xn .
(b) Give example of (xn) where all above inequalities
are strict. Hint; you may consider the following sequence
xn = 2n if n even and xn = 1 if n
odd.
. Suppose that the sequence (xn) satisfies
|xn –α| ≤ c |
xn-1- α|2 for all n.
Show by induction that c | xn- α| ≤ c |
x0 - α|2n , and give some condition
That is sufficient for the convergence
of (xn) to α.
Use part a) to estimate the number of iterations needed to
reach accuracy
|xn –α| < 10-12 in case c = 10 and
|x0 –α |= 0.09.
6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1)
random variables. Let M = max(X1,...,Xn).
(a) Find the density function of M. (b) Find E[M] and V[M].
Let X1,...,Xn be independent random
variables,and let X=X1+...+Xn be their
sum.
1. Suppose that each Xi is geometric with respective
parameter pi. It is known that the mean of X is equal to
μ, where μ > 0. Show that the variance of X is minimized if the
pi's are all equal to n/μ.
2. Suppose that each Xi is Bernoulli with respective
parameter pi. It is known that the mean of X is equal to
μ, where μ >...