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statistics on PlanetMath (Topic)

This entry is supposed to become a textbook on statistics - or at least a guide to the statistics-related entries on PlanetMath. It will refer to related entries (e.g. from probability) that are not listed in the same MSC.

(This is currently a stub article. See also Textbook projects on PlanetMath.)



"statistics on PlanetMath" is owned by PrimeFan. [ full author list (2) | owner history (1) ]
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Cross-references: Textbook projects on PlanetMath, PlanetMath, statistics
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This is version 4 of statistics on PlanetMath, born on 2004-11-13, modified 2006-11-27.
Object id is 6474, canonical name is StatisticsOnPlanetMath.
Accessed 6897 times total.

Classification:
AMS MSC: 62-01 (Statistics :: Instructional exposition )
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Matrix by PARASHAR on 2008-05-07 05:26:35
There is a 20x9 state table. Every cell determines its value (0 or 1) based on its eight neighbors: a cell having value 0 will change its value to 1 if and only if exactly three of its neighbors are having value 1; a cell having value 1 will remain to 1 if and only if two or three of its neighbors are having value 1.
All the changes are done simultaneously.
Find a possible state of the table whose next state contains 1s at following positions and zeros at rest of the positions.
[3,3], [3,4], [3,5], [3,8], [3,9], [3,10], [3,14], [3,18],
[4,9], [4,11], [4,11], [4,14], [4,15], [4,17], [4,18],
[5,4], [4,9], [4,10], [4,14], [4,16], [4,18],
[6,4], [6,9], [6,11], [6,14], [6,18],
[7,3], [7,4], [7,5], [7,8], [7,9], [7,10], [7,14], [7,18]
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March Ponder This by PARASHAR on 2008-03-03 22:44:02
S, has a good, G, which he is considering selling to B. The values of G to S and B are independent uniformly distributed random variables between 0,1. S and B know this and know their own valuations but not the valuation of the other party.

1. Suppose B makes a single offer which S accepts or rejects depending on whether or not the offer exceeds the value S places on the item. What should B offer to maximize his expected gain (the difference when a sale occurs between B's valuation of G and the sales price, 0 when no sale occurs) as a function of B's valuation of G? What is B's expected gain? What is S's?

2. Suppose there are 2 buyers B1, B2 (each with a valuation independently uniformly distributed between 0 and 1). Suppose each makes a single bid for G and S accepts the larger if and only if it exceeds S's valuation. Find a bidding strategy for B1 and B2 which is optimal in that if one adopts it the other can do no better than to adopt it also. What will be the expected gains of B1, B2 and S?

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January Ponder this by PARASHAR on 2008-01-02 21:40:41
The most easy ponder problem till now is here:

This month's puzzle concerns drawing balls from an urn. You know the urn contains 3 balls. You know each ball is white or black and that white balls are worth 1 unit, black balls are worth nothing. You know the urn is equally likely to contain 0,1,2 or 3 white balls. You have an opportunity to buy balls from the urn. Each ball costs c units (0<c<1). When you buy a ball one of the remaining balls in the urn is randomly selected and given to you. You may stop buying balls at any time based on the color of the balls you have received. Assuming you adopt the best strategy, what is your expected return as a function of c?


Is anyone interested
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Perfect solution for your business statistics needs by ajayi on 2007-12-07 11:16:20
I found this website that's been really productive for me, and feel the need to introduce this site to anyone that is interested in Warehouse Designing, and E-Commerce.


Link: http://www.thecyberprofessor.com
[ reply | up ]
December Ponder by PARASHAR on 2007-12-03 21:39:17
Suppose we have a large number, n, of parts and know at least one is defective. If we can test any number of parts at once (learning whether they are all good or at least one is defective) then it is well known that a binary search will identify a defective part after log2(n) tests.

This month's puzzle concerns a variation on this problem in which the defective part only fails intermittently. Assume when the bad part is tested (by itself or with other parts) it fails half the time and passes half the time. The following questions concern how many tests on average it takes various algorithms to identify a single defective part out of a large number n.

Suppose we have a large number, n, of parts and know at least one is defective. If we can test any number of parts at once (learning whether they are all good or at least one is defective) then it is well known that a binary search will identify a defective part after log2(n) tests.

This month's puzzle concerns a variation on this problem in which the defective part only fails intermittently. Assume when the bad part is tested (by itself or with other parts) it fails half the time and passes half the time. The following questions concern how many tests on average it takes various algorithms to identify a single defective part out of a large number n.

You should ignore any lower order terms arising from the fact that you can only test integral numbers of parts at once. Express the answers as C*log2(n) where C is a constant given to 4 decimal places (ie 1.0000 for the original problem).


1) Suppose we use the following algorithm. Select a subgroup of size .5*n at random and test it. If it passes repeat with different random subgroups of size .5*n until you find a subgroup which fails. Then recursively apply the algorithm to smaller groups until you have isolated the bad part. How many tests on average will this take?
2) Use the algorithm in 1) except you test random subgroups of size a*n instead of .5*n. What is the optimal value of a and how many tests on average will be required?
3) Divide the parts at random into 2 equal subgroups and test them alternatively until one fails. Then apply the algorithm recursively. How many tests on average will be required to isolate the bad part.
4) Same as 3) except at each stage the parts are divided into 3 equal subgroups which are tested in turn until one fails. Again how many tests on average will be needed to find the bad part?
5) (Not required for credit) The algorithm in 4 is pretty good but not optimal. How many tests does the optimal algorithm require?
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