then where

is the indicator function for the event that

lies in the image of the function. The

are Bernoulli variables with mean approximately

but are only approximately independent. I would have to

swot up on my probability theory to see if there’s a version

of the central limit theorem liberal enough to apply here,

but it does seem plausible that asymptotic normality holds. ]]>

The probability that the image has size k is

where is a Stirling number of of the second kind. The

mean size of an image is exactly

(n times the probability the image contains ).

Similar arguments give exact formulas for the variance etc.

The probability that the image has size k is $(n!/(n-k)!) S(n,k)/n^n$

where $S(n,k)$ is a Stirling number of of the second kind. The

mean size of an image is exactly $n(1-(n-1)^n/n^n)$

(n times the probability the image contains $1$).

Similar arguments give exact formulas for the variance etc.

12-17 for under 18

18 – 35 for youth

35 – 50 for middle age

51 to 60 for middle age

61 to 75 for old age ]]>