We need to define a bijective function
such that
.
Define
by
.
Then
is given by
.
Now,
Hence,
and
are isometric.
Problem Set 7 Functions between Metric Spaces 1. Construct a metric on (0,1) to make ((0.1), d) a complete metric space.
8. Let X (i-1,2) be independent N(0,1) random variables. a. Find the value of c such that P ( (X1 + X2 )2/( X2 -X1)2 < c ) =.90 b. Find P(2 X1 -3 X2< 1.5) c. Find 95th percentile of the distribution of Y-2 X1 -3 X2
(7) Let (Xi.d) and (X2, d) be complete metric spaces. Suppose that Yi c Xl is dense in Xi ½ C X2 is dense in X2, and there exists an isometry f: Y! → Ya, where Yi, ½ are endowed with the corresponding subspace metrics. Prove that there exists an isometry F: X1 → X2-
please explian every thing
consider two probability spaces ([0,1], β,A) and ([0,1], β,P), where Piand P2 are defined in terms of density function integrals as P (A) 20īdλ and乃(wi a-4 2-1 ,判 For A = {(wi, wa) C [0, 1] × [0, 1] I, das wi), compute (Rx P) (A)
consider two probability spaces ([0,1], β,A) and ([0,1], β,P), where Piand P2 are defined in terms of density function integrals as P (A) 20īdλ and乃(wi a-4 2-1 ,判 For A...
Let A =?15,?11 , B =?11, ??15 , C = (1,2), and D = (7, 6). Is there an isometry that transforms segment AB onto segment CD? Explain
Prove:
By taking the following problem as being given/true :
(Analysis on Metric Spaces)
Let f : [0, 1] x [0, 1] + R be defined by f(x,y) = ſi if y=x? if y #r? Show that f is integrable on [0, 1] x [0,1]. Let f : [0, 1] + R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that -y<= f(x) - f(y)< € for every I, Y E (0,1). The...
Show the following statements.
(b) on (0,00) <RXR (c) The interval (0,1) is equivalent to the interval (1,2].
Question 8 Lineal spaces between the pits in a CD-ROM are called: circles. segments. tracks. lands
Problem 2. If XN(0,1), determine the number of expected outcomes between 1 and 3 out of 10000 realizations of X. Next, run a computer simulation to carry out this experiment. How many outcomes fall within the interval?
Problem 8. Let n 2 2. Consider the following optimization problem min n(n-1) 1 i=1 Show that (0,1/(n 11/(n 1) is an optimal solution SolutionType your solution here.]
Problem 8. Let n 2 2. Consider the following optimization problem min n(n-1) 1 i=1 Show that (0,1/(n 11/(n 1) is an optimal solution SolutionType your solution here.]