#######################################################################
#
# Evolution of the U.S. income distribution, as a log-normal
# distribution, (from mean-variance analysis of US nominal GDP per
# capita):
#
#######################################################################
#
# Integrating the U.S. Income Distribution, 2010, (from U.S. Census
# Survey of Household data) to about the mean value, (the empirical
# annual mean income per capita is $39,791, the median is $19,658,
# and, the mode is $4,798, from ~/us.income.distribution.gp, which is
# in real dollars):
#
# From:
#
#     http://en.wikipedia.org/wiki/Log-normal_distribution
#
# Mean = exp (u + (s^2 / 2))
# Median = exp (u)
# Mode = exp (u - s^2)
#
# Where u is the offset of the log-normal distribution, and s the
# standard deviation, and obtaining the cumulative of the
# us.income.distribution file.
#
tsintegrate -t us.income.distribution | tail -1
87013.84376198585731153791      0.000504
#
# The cumulative of the us.income.distribution:
#
tsintegrate -t us.income.distribution | tsmath -t -d 0.000504
966.82263473903116797524        0.035714
    .
    .
    .
39639.53465977333008075120      0.839286
    .
    .
    .
87013.84376198585731153791      1.000000
#
# Or, empirically, 83.9% of incomes are less than the mean, and, the
# mean is about the same for the empirical and theoretical value.
#
# And, in calc(1), the cumulative to the mean for the theoretical:
#
; u = ln (19658)
; s = sqrt (2 * (ln (39791) - u))
; (1/2) + (1/2) * erf ((ln (39791) - u) / (sqrt (2 * (s^2))))
        0.72365059433417996312
#
# Or, theoretically, about 72.3% of incomes should be less than the
# mean; meaning the empirical has 11.6% more of the population than
# the theoretical.
#
# And, the empirical cumulative of incomes greater than
# $87013.84376198585731153791 is 0.000009, (i.e., about 1 in 100,000).
#
# From calc(1):
#
; (1/2) + (1/2) * erf ((ln (87013.84376198585731153791) - u) / \
      (sqrt (2 * (s^2))))
        0.89479911708583191039
#
# which is a theoretical excess of the cumulative of about 11%,
# compared with an empirical cumulative that is insignificant; i.e.,
# the difference between the theoretical and empirical to the mean is
# almost the same as the difference between the theoretical and
# empirical beyond the mean, with different signs.
#
#######################################################################
#
# The mean-variance analysis of the evolution of the U.S. Income
# Distribution, 1790 to 2010:
#
# (Note that the 2010 income distribution is in nominal 2010 US
# dollars, so nominal dollar history of the US GDP per capita must be
# used; additionally, the median-to calculate the gain-of US GDP per
# capita from 1790 to 2010 is unknown, meaning that the ratios, of
# both the theoretical and empirical values of the mean and the median
# will be compared for the evolution of the US GDP per capita.)
#
tsfraction us.nominal.gdp.capita.1790-2010 | tsrms -p
0.085756
#
tsfraction us.nominal.gdp.capita.1790-2010 | tsavg -p
0.034815
#
# US GDP per capita variance:
#
; 0.085756^2 - 0.034815^2
        0.006142007311
#
# Time interval:
#
; 2010 - 1790
        220
#
# Log-normal standard deviation, s:
#
; sqrt (0.006142007311) * sqrt (220)
        1.16242918426027140075
#
# Mean / Median, exp (s^2 / 2):
#
; exp (1.16242918426027140075^2 / 2)
        1.96525263445529821419
#
# as a theoretical number. As an empirical number:
#
; 39791 / 19658
        2.02416319055855122596
#
# Mode / Median = exp (- s^2):
#
; exp (- 1.16242918426027140075^2)
        0.25891858549408337365
#
# as a theoretical number. As an empirical number:
#
; 4798 / 19658
        0.24407365957879743616