ABCDEFGHIJKLMNOPQRSTUVWXYZ
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Unexpected Extra Expense Calculation for Each 1-Year Period
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Exponential Decay Function
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y=A * e^(c*x)
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A =40,000.00
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c =-20.00
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Loss in Period
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xy
Cumulative frequency for each of 500 LuckLand households in 50 years
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0.00 40,000.000.0
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0.000139,920.080.0
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0.000239,840.320.0
Area under the curve from 0 to 1 =
Average loss above a threshold B
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0.000539,601.990.1
Integral of y from 0 to 1
= 1/c * ( B - A - B*ln(B/A) )
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0.01 32,749.231.0
= A * 1/c * e^(c) - A * 1/c *e^0
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0.02 26,812.802.0
= A/c * ( e^(c) - 1)
If B =100.00
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0.03 21,952.473.02,000.00
Average Loss Above B =
1,965.04
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0.04 17,973.164.0
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0.05 14,715.185.0
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0.0758,925.217.5Average
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0.10 5,413.4110.0
Area under the curve from 0.55 to 1 =
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0.15 1,991.4815.0
= A/C *( (e^c - e^(0.55*c) )
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0.20 732.6320.00.030.07
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0.25 269.5225.0
In each year, 45% of the households have expenses less than a dollar (truncated to \$0).
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0.30 99.1530.0
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0.35 36.4835.0
Area under the curve from 0.2 to 0.55 =
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0.40 13.4240.0
= A/C *( (e^(0.55*c) - e^(0.2*c) )
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0.45 4.9445.036.60104.57
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0.50 1.8250.0 Median
"In another 35% of the households, extra expenses are less than \$733 and the average is \$104.57."
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0.55 0.6755.0
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0.60 0.2560.0
Area under the curve from 0.1 to 0.2 =
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0.65 0.0965.0
= A/C *( (e^(0.2*c) - e^(0.1*c) )
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0.70 0.0370.0234.042,340.39
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0.75 0.0175.0
"In another 10% of the households, the average expense is \$2,340.39."
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0.80 0.0080.0
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0.85 0.0085.0
Area under the curve from 0 to 0.10 =
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0.90 0.0090.0
= A/C *( (e^(0.1*c) - 1) )
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0.95 0.0095.01,729.3317,293.29
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1.00 0.00100.0
"But in the last 10% of the households, the average extra expense is \$17,293.29."
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2,000.002,000.00
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"And over 50 years, each household is likely to have at least
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one extra expense of \$27,000 to \$40,000."
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Integrals
32,968.00
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http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions
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