Tables 4: Tests in banknotes' low places

Index

Table 4-1: Gradual D9F based on lognormal distribution
Table 4-2: Medium D9F based on lognormal distribution
Table 4-3: Steep D9F based on lognormal distribution
Table 4-4: Average D9F
Table 4-5: Test A2 of 1,3C
Table 4-6: Test A2 results
Table 4-7: Test B2 of 1,6C
Table 4-8: Test B2 results
Table 4-9: D9F Distribution of 100,000 random numbers
Table 4-10: Test C2 results
Table 4-11: Test D2 results
Table 4-12: NBCs of test D2
Table 4-13: Test E2 results
Table 4-14: NBCs of test E2
Table 4-15: Performances of 8 C2Ds in BLP

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Table 4-1

Gradual D9F (Decreasing 9 Frequencies) based on lognormal distribution

Num.	Calculation	Total	Ration
1 2 3 4 5 6 7 8 9 10	f(0.1)+f(1.1)+...+f(19.1) f(0.2)+f(1.2)+...+f(19.2) f(0.3)+f(1.3)+...+f(19.3) f(0.4)+f(1.4)+...+f(19.4) f(0.5)+f(1.5)+...+f(19.5) f(0.6)+f(1.6)+...+f(19.6) f(0.7)+f(1.7)+...+f(19.7) f(0.8)+f(1.8)+...+f(19.8) f(0.9)+f(1.9)+...+f(19.9) f(1.0)+f(2.0)+...+f(20.0)	0.952 1.162 1.211 1.178 1.110 1.030 0.948 0.870 0.798 0.731	0.103 0.126 0.131 0.127 0.120 0.111 0.102 0.094 0.086 -
Total		9.990	1.000

¤ f(x)=1/x/√(2πe^(log(x))2). μ=0. σ=1.
¤ Mode=1/e=0.368. Median=1. Average=√e=1.649.
¤ f(20.0)=0.0002, and f(x) (20.0<x) is enough small to be disregarded in calculation.
¤ Ration of number 1 is 0.952/(9.990-0.731)=0.103.

Table 4-2

Medium D9F based on lognormal distribution

Num.	Calculation	Total	Ration
1 2 3 4 5 6 7 8 9 10	f(0.2)+f(2.2)+...+f(18.2) f(0.4)+f(2.4)+...+f(18.4) f(0.6)+f(2.6)+...+f(18.6) f(0.8)+f(2.8)+...+f(18.8) f(1.0)+f(3.0)+...+f(19.0) f(1.2)+f(3.2)+...+f(19.2) f(1.4)+f(3.4)+...+f(19.4) f(1.6)+f(3.6)+...+f(19.6) f(1.8)+f(3.8)+...+f(19.8) f(2.0)+f(4.0)+...+f(20.0)	0.736 0.820 0.728 0.613 0.511 0.426 0.357 0.302 0.257 0.220	0.155 0.173 0.153 0.129 0.107 0.090 0.075 0.064 0.054 -
Total		4.971	1.000

Table 4-3

Steep D9F based on lognormal distribution

Num.	Calculation	Total	Ration
1 2 3 4 5 6 7 8 9 10	f(0.4)+f(4.4)+...+f(16.4) f(0.8)+f(4.8)+...+f(16.8) f(1.2)+f(5.2)+...+f(17.2) f(1.6)+f(5.6)+...+f(17.6) f(2.0)+f(6.0)+...+f(18.0) f(2.4)+f(6.4)+...+f(18.4) f(2.8)+f(6.8)+...+f(18.8) f(3.2)+f(7.2)+...+f(19.2) f(3.6)+f(7.6)+...+f(19.6) f(4.0)+f(8.0)+...+f(20.0)	0.692 0.517 0.352 0.244 0.174 0.128 0.096 0.074 0.058 0.046	0.296 0.221 0.151 0.104 0.075 0.055 0.041 0.032 0.025 -
Total		2.382	1.000

Table 4-4

Average D9F

Num.	D9F			Avg. D9F
	Gradual	Medium	Steep
1 2 3 4 5 6 7 8 9	0.103 0.126 0.131 0.127 0.120 0.111 0.102 0.094 0.086	0.155 0.173 0.153 0.129 0.107 0.090 0.075 0.064 0.054	0.296 0.221 0.151 0.104 0.075 0.055 0.041 0.032 0.025	0.185 0.173 0.145 0.120 0.101 0.085 0.073 0.063 0.055
Total	1.000	1.000	1.000	1.000

¤ Average D9F of number 1 is (0.103+0.155+0.296)/3=0.185.
¤ Average D9F is used on tests in BLP (Banknotes' Low Places) where denominations are not restocked with.

Table 4-5

Test A2 of 1,3C: Total QB (Quantity of Banknotes) based on minimum NBCs (the Number of Banknotes a payer Carries) to make Y1 to Y9 without change, using 1-yen banknotes and 3-yen banknotes

To make	How	NBC		D9F	QB
		[3]	[1]
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 1+1 3 3+1 3+1+1 3+3 3+3+1 3+3+1+1 3+3+3	0 0 1 1 1 2 2 2 3	1 2 0 1 2 0 1 2 0	0.185 0.173 0.145 0.120 0.101 0.085 0.073 0.063 0.055	0.185 0.346 0.154 0.247 0.309 0.180 0.228 0.260 0.175
Total				1.000	2.083

¤ QB is the sum of NBC[1] (NBC of 1-yen banknotes) and 1.06 times NBC[3], each of which is multiplied by D9F. e.g. QB to make Y5 is (2+1.06)*0.101=0.309.
¤ Total NBC[1] with D9F is 1*0.185+2*0.173+...+0*0.055=1.052. And total NBC[3] with D9F is 0*0.185+...+3*0.055=0.973.

Table 4-6

Test A2 results: Total QBs and their Ex‰s, of 8 C2Ds

C2D	Total NBC		Total QB	Ex‰
	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1.706 0.973 0.615 0.377 0.276 0.191 0.118 0.055	0.559 1.052 1.511 2.086 2.315 2.634 3.027 3.476	2.367 2.083 2.163 2.486 2.608 2.836 3.152 3.534	136 0 38 193 252 361 513 696

¤ QB is the sum of NBC[1] and 1.06 times NBC[N].

Table 4-7

Test B2 of 1,6C: The total of minimum LCs (Load of Count in cash transactions) to make Y1 to Y9 without change, using 1-yen banknotes and 6-yen banknotes

To make	How	NBM		D9F	LC
		[6]	[1]
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9	1 1+1 1+1+1 1+1+1+1 1+1+1+1+1 6 6+1 6+1+1 6+1+1+1	0 0 0 0 0 1 1 1 1	1 2 3 4 5 0 1 2 3	0.185 0.173 0.145 0.120 0.101 0.085 0.073 0.063 0.055	0.185 0.346 0.435 0.480 0.505 0.128 0.183 0.221 0.248
Total				1.000	2.729

¤ LC is the sum of NBM[1] (the Number of 1-yen Banknotes Moved in cash transactions) and 1.5 times NBM[6], each of which is multiplied by D9F. e.g. LC to make Y8 is (2+1.5)*0.063=0.221.
¤ Total NBM[1] with D9F is 1*0.185+2*0.173+...+3*0.055=2.315. And total NBM[6] with D9F is 0*0.185+...+1*0.055=0.276.

Table 4-8

Test B2 results: Total LCs and their Ex‰s, of 8 C2Ds

C2D	Total NBM		Total LC	Ex‰
	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	1.706 0.973 0.615 0.377 0.276 0.191 0.118 0.055	0.559 1.052 1.511 2.086 2.315 2.634 3.027 3.476	3.118 2.512 2.434 2.652 2.729 2.921 3.204 3.559	281 32 0 90 121 200 317 462

¤ LC is the sum of NBM[1] and 1.5 times NBM[N].

Table 4-9

D9F Distribution of 100,000 random numbers

Num.	Expectation	Distribution
1 2 3 4 5 6 7 8 9	18,500 17,300 14,500 12,000 10,100 8,500 7,300 6,300 5,500	18,495 17,300 14,362 11,955 10,116 8,559 7,369 6,290 5,554
Total	100,000	100,000

Table 4-10

Test C2 results: Average QBs and their Ex‰s, of 8 C2Ds

C2D	Maximum NBC			Average NBC		Avg. QB	Ex‰
	[N]	[1]	Tot.	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	4 3 2 1 1 2 3 4	1 2 3 4 5 3 3 5	5 4 4 5 5 4 4 5	1.995 1.197 0.799 0.498 0.399 0.895 1.192 0.993	0.500 0.901 1.295 1.999 2.098 1.202 0.901 1.496	2.615 2.169 2.142 2.527 2.521 2.150 2.164 2.549	221 13 0 180 177 4 10 190

¤ QB is the sum of NBC[1] and 1.06 times NBC[N].

Table 4-11

Test D2 results: Average LCs and their Ex‰s, of 8 C2Ds

C2D	Proportion		Average NBM		Avg. LC	Ex‰
	P>9	C>9	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	0.398 0.398 0.398 0.398 0.497 0.440 0.375 0.260	0 0 0 0 0 0 0.099 0.099	1.296 0.811 0.683 0.445 0.582 0.690 0.741 0.653	0.578 1.050 1.180 1.618 1.232 1.034 1.365 2.147	2.919 2.665 2.603 2.683 2.603 2.509 2.951 3.485	164 62 38 70 38 0 176 389

¤ LC is the sum of NBM[1], 1.5 times NBM[N], and DD (Double Digit value). e.g. LC of 1,9C is 2.147+1.5*0.653+0.260+0.099=3.485.
¤ DD is the sum of P>9 (pay exceeding Y9) and C>9 (change exceeding Y9).

Table 4-12

NBCs of test D2

C2D	Maximum NBC			Average NBC
	[N]	[1]	Total	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	22 12 8 1 14 105 9,483 92	9 19 18 28 18 26 28 32	23 21 19 29 19 110 9,487 98	3.224 1.305 1.013 0.468 1.725 22.628 4,662.5 15.348	1.044 1.867 2.143 3.226 2.225 2.557 2.399 3.515

¤ NBC[8] of 1,8C proliferates as transactions go on. Other C2Ds let a payer carry dozens of banknotes at times.

Table 4-13

Test E2 results: Average LCs and their Ex‰s, of 8 C2Ds

C2D	Proportion		Average NBM		Avg. LC	Ex‰
	P>9	C>9	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	0.398 0.398 0.398 0.398 0.497 0.474 0.406 0.324	0 0 0 0 0 0 0.086 0.100	1.341 0.820 0.691 0.461 0.591 0.689 0.755 0.671	0.580 1.049 1.182 1.659 1.241 1.138 1.581 2.447	2.989 2.676 2.617 2.748 2.625 2.646 3.205 3.877	142 23 0 50 3 11 225 481

¤ Each of average LC is somewhat larger than LC of test D2 (table 4-11).

Table 4-14

NBCs of test E2

C2D	Maximum NBC			Average NBC
	[N]	[1]	Total	[N]	[1]
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	7 7 6 1 6 6 7 8	6 7 7 8 8 9 9 9	12 12 11 9 13 13 13 14	2.606 1.295 1.023 0.473 1.662 2.592 2.833 2.108	1.030 1.769 2.012 2.688 2.073 2.175 2.054 2.796

¤ NBCs are under control.

Table 4-15

Performances of 8 C2Ds in BLP

C2D	Ex‰ of test				Total Ex‰
	A2 0.05	B2 0.05	C2 0.45	E2 0.45
1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9	136 0 38 193 252 361 513 696	281 32 0 90 121 200 317 462	221 13 0 180 177 4 10 190	142 23 0 50 3 11 225 481	184 18 2 118 99 35 147 360

¤ Ex‰ of tests A2, B2, C2, and E2 are mixed in the ratio of 5:5:45:45 to produce the total Ex‰. e.g. Total Ex‰ of 1,2C is 0.05*136+0.05*281+0.45*221+0.45*142=184.

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1¤4

An Efficient Combination of the Denominations of a Currency
 Tables 1: Measurements of coins
 Tables 2: Tests in coins' places
 Tables 3: Measurements of banknotes
 Tables 4: Tests in banknotes' low places
 Tables 5: Tests in banknotes' high places
 Tables 6: Tests of C3Ds

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© 2004 Takashi Shimazaki
Updated: April 2, 2014