Expression 11: "t" left parenthesis, "i" , right parenthesis equals left brace, 1 less than or equal to "i" less than floor left parenthesis, StartFraction, "k" Over "S" , EndFraction , right parenthesis : StartFraction, "S" Over "i" "k" , EndFraction , "i" equals floor left parenthesis, StartFraction, "k" Over "S" , EndFraction , right parenthesis : StartFraction, "S" ln left parenthesis, StartNestedFraction, "S" NestedOver "E" , EndNestedFraction , right parenthesis Over "k" , EndFraction , "i" greater than floor left parenthesis, StartFraction, "k" Over "S" , EndFraction , right parenthesis : 0 , right braceti=1≤i<floorkS:Sik,i=floorkS:SlnSEk,i>floorkS:0
11
B is used to normalize the robust distribution.
12
Expression 13: "B" equals Start sum from "i" equals 1 to "k" , end sum, left parenthesis, "p" left parenthesis, "i" , right parenthesis plus "t" left parenthesis, "i" , right parenthesis , right parenthesis equals 1.4 2 8 9 6 9 5 2 6 4 8B=k∑i=1pi+ti
equals=
1.4 2 8 9 6 9 5 2 6 4 81.42896952648
13
u is the robust distribution.
14
Expression 15: "u" left parenthesis, "i" , right parenthesis equals left brace, 1 less than or equal to "i" less than or equal to "k" : StartFraction, left parenthesis, "p" left parenthesis, "i" , right parenthesis plus "t" left parenthesis, "i" , right parenthesis , right parenthesis Over "B" , EndFraction , right brace Has graph. To audio trace, press ALT+T.ui=1≤i≤k:pi+tiB
15
Compare data points (note the change in 1 and 2, as well as the spike later on)
16
"i"i
"p" left parenthesis, "i" , right parenthesis Has graph. To audio trace, press ALT+T.pi
"u" left parenthesis, "i" , right parenthesis Has graph. To audio trace, press ALT+T.ui
11
r1c2: 0.0 6 2 50.0625
r1c3: 0.1 1 9 5 2 9 6 60.11952966
r1c4:
22
r2c2: 0.50.5
r2c3: 0.3 8 7 7 9 8 4 20.38779842
r2c4:
33
r3c2: 0.1 6 6 6 6 6 6 70.16666667
r3c3: 0.1 4 1 8 9 8 1 20.14189812
r3c4:
44
r4c2: 0.0 8 3 3 3 3 3 3 30.083333333
r4c3: 0.0 7 7 2 6 5 0 4 50.077265045
r4c4:
55
r5c2: 0.0 50.05
r5c3: 0.0 5 0 1 4 8 6 20.05014862
r5c4:
66
r6c2: 0.0 3 3 3 3 3 3 3 30.033333333
r6c3: 0.0 3 5 9 5 8 8 0 80.035958808
r6c4:
77
r7c2: 0.0 2 3 8 0 9 5 2 40.023809524
r7c3: 0.0 2 7 4 8 9 4 3 10.027489431
r7c4:
88
r8c2: 0.0 1 7 8 5 7 1 4 30.017857143
r8c3: 0.0 2 1 9 7 0 4 9 90.021970499
r8c4:
99
r9c2: 0.0 1 3 8 8 8 8 8 90.013888889
r9c3: 0.1 0 3 9 2 3 10.1039231
r9c4:
1010
r10c2: 0.0 1 1 1 1 1 1 1 10.011111111
r10c3: 0.0 0 7 7 7 5 6 1 10.007775611
r10c4:
1111
r11c2: 0.0 0 9 0 9 0 9 0 9 10.0090909091
r11c3: 0.0 0 6 3 6 1 8 6 3 50.0063618635
r11c4:
1212
r12c2: 0.0 0 7 5 7 5 7 5 7 60.0075757576
r12c3: 0.0 0 5 3 0 1 5 5 2 90.0053015529
r12c4:
1313
r13c2: 0.0 0 6 4 1 0 2 5 6 40.0064102564
r13c3: 0.0 0 4 4 8 5 9 2 9 40.0044859294
r13c4:
1414
r14c2: 0.0 0 5 4 9 4 5 0 5 50.0054945055
r14c3: 0.0 0 3 8 4 5 0 8 2 30.0038450823
r14c4:
1515
r15c2: 0.0 0 4 7 6 1 9 0 4 80.0047619048
r15c3: 0.0 0 3 3 3 2 4 0 4 70.0033324047
r15c4:
1616
r16c2: 0.0 0 4 1 6 6 6 6 6 70.0041666667
r16c3: 0.0 0 2 9 1 5 8 5 4 10.0029158541
r16c4:
r17c2:
r17c3:
r17c4:
17
Check if distributions are normalized.
18
Expression 19: Start sum from "n" equals 1 to "k" , end sum, "p" left parenthesis, "n" , right parenthesis equals 1k∑n=1pn
equals=
11
19
Expression 20: Start sum from "n" equals 1 to "k" , end sum, "u" left parenthesis, "n" , right parenthesis equals 1k∑n=1un