[R] family question
Douglas Bates
bates at stat.wisc.edu
Wed Aug 30 20:44:00 CEST 2000
Troels Ring <tring at mail1.stofanet.dk> writes:
> Dear friends. Please see the program below and answer if it does simulate a
> population of 1.000.000 families, each with at max 20000 children (typical
> in Denmark, you know), constructed such that each family stops having
> children when more boys than girls are present ? Equal numbers of boys and
> girls are got in the population, according to the simulation, is that obvious ?
I am not an expert in probability or in stochastic processes so I
can't say that it is "obvious". I can say it is not unreasonable if
you consider the sizes of families. Basically what happens is that
large families have nearly equal numbers of girls and boys and also
have a high weight in the calculation of the population proportion.
(You can only get odd numbered family sizes according to your rule and
for a family of size 2K + 1 there will be K girls and K + 1 boys.)
>From simulation it seems that there is a non-negligible probability of
very large family sizes. If you get a high proportion of girls early
then it can take a long time for the number of boys to catch up.
Most of the families have only 1 or 3 children but about 1% or
2% of the time you get families of size 500 or more.
In fact when I simulated 1000 families with a maximum allowable family size of
999, I got 23 families that would have had more than 999 children.
> famsz <- double(1000)
> ind <- 1:1000
> for (i in seq(along = famsz)) famsz[i] <- min(ind[cumsum(ifelse(runif(1000) > 0.5, 1, 0)) > (ind/2)])
There were 23 warnings (use warnings() to see them)
> table(famsz)
famsz
1 3 5 7 9 11 13
497 141 62 25 31 24 17
15 17 19 21 23 25 27
15 9 6 12 8 10 6
29 31 33 35 37 39 41
2 6 7 2 2 3 3
43 45 47 51 55 57 59
2 1 4 3 5 2 1
61 63 65 69 71 73 75
2 3 1 1 1 1 1
77 91 97 99 103 107 109
5 1 2 3 1 2 1
111 113 121 127 137 139 143
1 1 1 1 2 1 1
147 163 167 169 173 177 191
1 1 1 3 1 1 2
215 235 243 251 275 279 281
1 1 1 1 1 1 1
293 307 321 359 401 411 421
1 1 1 1 1 1 1
459 467 489 537 573 581 603
1 1 1 1 1 1 1
629 631 661 771 839 863 929
1 1 1 1 1 1 1
2147483647
23
> ND <- NP <- NULL #ND - number boys, P: girls
> for (j in 1:1000000) # number of families
> {
> n1 <- n2 <- NULL
> for (i in 1:20000) {
> n1[i] <- rbinom(1,1,0.5) # each equally likely - here number of boys
> n2[i] <- 1- n1[i] # and girls
> if(sum(n1)>sum(n2)) break
> }
> ND[j] <- sum(n1)
> NP[j] <- sum(n2)
> }
> sum(ND)/sum(NP)
> j
>
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--
Douglas Bates bates at stat.wisc.edu
Statistics Department 608/262-2598
University of Wisconsin - Madison http://www.stat.wisc.edu/~bates/
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