This came up due to a question from Anders Hald:
Bernoulli calculated an approximation to the smallest n so that
P(0.58 <= x/n <= 0.62) >= 1000/1001
What is the exact value?
Now try
n <- 6350:6500
Pr <- function(n)pbinom(0.62*n,n,0.6) -
pbinom(0.58*n,n,0.6) + dbinom(0.58*n,n,0.6)
plot(n,Pr(n),type="b")
abline(h=1000/1001)
min(n[Pr(n)>1000/1001])
Next, try
Pr <- function(n)pbinom(0.62*n,n,0.6) -
pbinom(0.58*n + 1e-7,n,0.6) + dbinom(0.58*n,n,0.6)
plot(n,Pr(n),type="b")
abline(h=1000/1001)
min(n[Pr(n)>1000/1001])
At least on Linux, you get quite different results.
The reason is of course that
> 0.58*6400-3712
[1] -4.547474e-13
and dbinom has a fuzz factor, but pbinom unceremoniously takes
floor(x). I suppose we ought to build the same fuzz into pbinom?
--
O__ ---- Peter Dalgaard Blegdamsvej 3
c/ /'_ --- Dept. of Biostatistics 2200 Cph. N
(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard@biostat.ku.dk) FAX: (+45) 35327907
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