Statistic problem with a hypercube ?

[v***@yahoo.fr]

Hi,

Sorry but I am reading a book (The Elements of Statistical Learning) and there is this question. I really don't understand what to do.
So any help would be very appreciated.

A hypercube with side length 1 in d dimensions is defined to be the set of points (x1, x2, ..., xd) such that 0≤ xj ≤1 for all j = 1, 2, ..., d. Define the boundary region of the hypercube to be the set of all points such that there exists a j for which 0≤ xj ≤.05 or .95≤ xj ≤1 (i.e., it is the set of all points that have at least one dimension in the most extreme 10% of possible values). What proportion of the volume of a hypercube of dimension 50 is in the boundary region ?

Best,

[Gordon Sande]

Post by v***@yahoo.fr
Hi,
Sorry but I am reading a book (The Elements of Statistical Learning)
and there is this question. I really don't understand what to do.
So any help would be very appreciated.
A hypercube with side length 1 in d dimensions is defined to be the set
of points (x1, x2, ..., xd) such that 0≤ xj ≤1 for all j = 1, 2, ...,
d. Define the boundary region of the hypercube to be the set of all
points such that there exists a j for which 0≤ xj ≤.05 or .95≤ xj ≤1
(i.e., it is the set of all points that have at least one dimension in
the most extreme 10% of possible values). What proportion of the volume
of a hypercube of dimension 50 is in the boundary region ?
Best,

Change the way you look at the problem. Ask about the interior. What is
the proportion
that is in the interior in any one dimension. Now look at the
definition of boundary
region and relate it to how many dimensions have to be in the interior.
I leave the
arithmetic to the original poster.

A long time ago I knew several folks who did number theory and packing
problems. They
claimed that once the dimension got up a bit (4 and up) that the unit
sphere was better
imaged as a thing that looked more like a child's jack than the usual
low dimension
round ball. Question: Why did I mention this obscure topic given the above?

[v***@yahoo.fr]

Post by Gordon Sande

Post by v***@yahoo.fr
Hi,
Sorry but I am reading a book (The Elements of Statistical Learning)
and there is this question. I really don't understand what to do.
So any help would be very appreciated.
A hypercube with side length 1 in d dimensions is defined to be the set
of points (x1, x2, ..., xd) such that 0≤ xj ≤1 for all j = 1, 2, ...,
d. Define the boundary region of the hypercube to be the set of all
points such that there exists a j for which 0≤ xj ≤.05 or .95≤ xj ≤1
(i.e., it is the set of all points that have at least one dimension in
the most extreme 10% of possible values). What proportion of the volume
of a hypercube of dimension 50 is in the boundary region ?
Best,

Change the way you look at the problem. Ask about the interior. What is
the proportion
that is in the interior in any one dimension. Now look at the
definition of boundary
region and relate it to how many dimensions have to be in the interior.
I leave the
arithmetic to the original poster.
A long time ago I knew several folks who did number theory and packing
problems. They
claimed that once the dimension got up a bit (4 and up) that the unit
sphere was better
imaged as a thing that looked more like a child's jack than the usual
low dimension
round ball. Question: Why did I mention this obscure topic given the above?

Hi,
I am not sure to have understood everything, but anyway I think I got the essential.
So many thanks

Best,

[n***@gmail.com]

We know that the volume of the whole hypercube is 150 = 1. The volume of the interior of the hypercube is 0.950 = 0.005. Thus, the volume of the boundary is 1-0.005 = 0.995.

[n***@gmail.com]

We know that the volume of the whole hypercube is 1^50 = 1. The volume of the interior of the hypercube is 0.9^50 = 0.005. Thus, the volume of the boundary is 1-0.005 = 0.995.

hope this s helpful

[s***@gmail.com]

when i read even i was confused that do we need to calculate the volume of boundary region or volume with in the boundary region.But if you read the question carefully you will get the gist that it is asking volume of boundary region.Therefore we need to subtract the volume of interior region from the total volume.

Hope that helps.

[Rich Ulrich]

Post by s***@gmail.com
when i read even i was confused that do we need to calculate the volume of boundary region or volume with in the boundary region.But if you read the question carefully you will get the gist that it is asking volume of boundary region.Therefore we need to subtract the volume of interior region from the total volume.
Hope that helps.

The original problem, in 2014, was answered more completely
back in 2014. See --

https://groups.google.com/forum/#!topic/sci.stat.consult/Mi8bAxSQzjs

The original --

Post by s***@gmail.com
Hi,
Sorry but I am reading a book (The Elements of Statistical Learning) and there is this question. I really don't understand what to do.
So any help would be very appreciated.
A hypercube with side length 1 in d dimensions is defined to be the set of points (x1, x2, ..., xd) such that 0? xj ?1 for all j = 1, 2, ..., d. Define the boundary region of the hypercube to be the set of all points such that there exists a j for which 0? xj ?.05 or .95? xj ?1 (i.e., it is the set of all points that have at least one dimension in the most extreme 10% of possible values). What proportion of the volume of a hypercube of dimension 50 is in the boundary region ?