Math Is Fun Forum

statistically_challenged

Member

Registered: 2006-11-05

Posts: 1

A reversal of the Birthday Problem

I hope today finds you well and in good spirits.

I'm trying to develope a statistical model and haven't done so in years. Any help is appreciated.

My question goes something like this:

My question is what are the odds of blood relatives dying on a personally or globally significant date (See below).

We observe two calendars in the family, Gregorian (Greek Orthodox) for religious purposes, and Julian (Episcopalian).

Helene, Jean, and Xenia are my aunts. Estelle and Edward are my mother and father.

****Name**********Birth *******Death**Age**Date Sign.*Calendar
1999 Helene---01 Oct 1911--09 Jan 1999-88--Christmas---Gregorian

2000 Jean*----07 Sep 1913--26 Dec 2000-87--Christmas---Julian

2001 Estelle--31 May 1925--28 Dec 2001-76--Christmas---Julian

2002 None

2003 None

2004 None

2005 Xenia----31 Dec 1917--11 Sep 2005-88-9/11--------Julian

2006 Edward--15 May 1921--07 Sep 2006-85-Sister Jean--Julian
--------------------------------------------Birthday*

Looking at the Birthday Problem:
To find the probability that both the second person and the third person
will have different birthdays (or, in my model, death days), we have to
multiply:

(365/365) * (364/365) * (363/365) which is about 99.18%.

Coincidences become more likely if we allow a little latitude into the
definition of what constitutes a coincidence - for example allowing
birthdays separated by no more than r days of each other to constitute a
“hit”. We can model this “nearmiss effect” using a balls-in-urns model; the
argument is somewhat more involved (see for example Naus, I. I. (1968). An
extension of the Birthday Problem, The American Statistician, 22, 27-29.)
and leads to:

P(>2 birthdays separated by <r days)
= 1— [(364 - rN)!365 to the power of l-N /(365 - (r + 1)N)!]

Life tables from various sources tell us, in very general terms, on
average, how long we can expect to live. True, predictors in life span can
be life style, diet, sleep patterns, genetics, etc.. I would prefer not
getting into that level of analysis since it would not be likely that such
a model will be available in our lifetimes. Further, medical science would
be doing back flips if that were possible. At this point in time, it simply
ISN'T possible. Therefore, lets leave that out so the smell of melting
circuitry does not begin to permeate the room.

I need to incorporate considerations for deaths occuring in a sequence of
years. I also need to incorporate considerations for near misses, that is,
variations from a date by, say, 3 days.

The sample size of blood relatives in the generational tier is 6. 5 members
in the tier bloodline have passed. The total sample size of this
generational tier is 11. I have chosen not to look at other deaths that
have occurred because a) they were in-laws, not in the blood line, and b)
had no readily identifiable significant date associated with their passing.

So can we start with:

1) Is the math in the birthday problem a BASIS for developing my model?

2) How do I account for near misses given a Greek Orthodox calender and
Episcopalian calender where Christmas is, as we all know is 12/25 and fewer
of us know is 01/07 (Greek Orthodox).

3)How do I account for different spans in years which target different date
clusters(i.e., 1999-2006, 1999-2001, 2005-2006, 2002-2005), if that is
appropriate.

I really need some solid input into this. Any help is appreciated.

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: A reversal of the Birthday Problem

Intersting. This post covers a lot of different topics in some detail and I am left unsure what exactly it is that you're trying to do. It sounds as though there are a lot of different aspects that you want to look at in the model and this can be overwhelming for potential responders, most of whom aren't experts in statistics and none of whom have probably have read the papers you're referring to.

It would help if you could reduce your post to a simpler question or two if there are some concepts you are struggling with. At this point I'm not sure what the level is of the marterial you're struggling with.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: A reversal of the Birthday Problem

most of whom aren't experts in statistics and none of whom have probably have read the papers you're referring to.
-he he, you are right.

However, this time I have been patient, and I think s-c you probably made the problem too complicated.

First, forget about birthday and life-span. Let's say a person could die at any date of any year. And generally, he/she has the same chance, or almost the same chance to die on some Jan 1st , some July 1st or some Dec 31st. In the same year the date may represent some different life-spans, but they may represent another different ones in different years.

Hence the problem could be simplier. Count how many significant days in a year, and how-many/365.25 would be the general frequency of significant days across the history, then mutiply each relative's how-many/365.25 to represent they all die on some significant day.

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X'(y-Xβ)=0