Coins pooling or not

Student ID: F74082125
Student Name: 張議隆
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TOC

• Coins pooling or not
  • How to run my code
  • Seperate trials
    • Possible outcomes of 4 trials(s1)
    • Possible outcomes of 5 trials(s2)
    • Probability table of s1, s2
    • Normalized form of above table
    • Simulation Output
  • Combined trials
    • Possible outcomes of 9 trials
    • Combination table
    • Probability Table
    • Simulation Output

How to run my code

Most commands are already written in Makefile

SEPERATE_CODE=Seperate_experiment
COMBINE_CODE=Combined_experiment

setup:
        export GSL_RNG_SEED=74082125
run_seperate:
        gcc -o $(SEPERATE_CODE)  $(SEPERATE_CODE).c  GSLfun.c -lgsl -lgslcblas -lm
        ./$(SEPERATE_CODE)

run_combine:
        gcc -o $(COMBINE_CODE)  $(COMBINE_CODE).c  GSLfun.c -lgsl -lgslcblas -lm
        ./$(COMBINE_CODE)

clean:
        rm $(SEPERATE_CODE)
        rm $(COMBINE_CODE)

To set the RNG seed, run make setup or make in short.
This command may not always work(at least in my enviorment). If make setup did not set the RNG seed, run export GSL_RNG_SEED=74082125 manually to set the seed.
Then you could run make run_seperate and run_combine to get the simulation result.
If the simulation took too long to generate the result, modify line 23 to a smaller number.

const unsigned long long int sample_count = 1000 * (1 << sample_power);

Seperate trials

Possible outcomes of 4 trials(s1)

Table entry

$(s<4)$ taken directly from given pdf file

$C_i^{i-\frac{1}{2}}={\displaystyle \frac{(i-\frac{1}{2})(i-\frac{3}{2})\cdots \frac{1}{2}}{i!}}$

s	0	1	2	3	4
$C_s^{s-0.5}$	1	$\frac{1}{2}$	$\frac{3}{8}$	$\frac{5}{16}$	$\frac{35}{128}$
$P[s]$	$\frac{35}{128}$	$\frac{5}{32}$	$\frac{9}{64}$	$\frac{5}{32}$	$\frac{35}{128}$
$128\ast P[s]$	$35$	$20$	$18$	$20$	$35$

Possible outcomes of 5 trials(s2)

Extended from the table above, and re-calculate

$P[s]$

s	0	1	2	3	4	5
$C_s^{s-0.5}$	1	$\frac{1}{2}$	$\frac{3}{8}$	$\frac{5}{16}$	$\frac{35}{128}$	$\frac{63}{256}$
$P[s]$	$\frac{63}{256}$	$\frac{35}{256}$	$\frac{15}{128}$	$\frac{15}{128}$	$\frac{35}{256}$	$\frac{63}{256}$
$256\ast P[s]$	$63$	$35$	$30$	$30$	$35$	$63$

Probability table of s1, s2

	$s_1=0$	1	2	3	4
$s_2=0$	$\frac{63}{256}\ast \frac{35}{128}$	$\frac{63}{256}\ast \frac{5}{32}$	$\frac{63}{256}\ast \frac{9}{64}$	$\frac{63}{256}\ast \frac{5}{32}$	$\frac{63}{256}\ast \frac{35}{128}$
1	$\frac{35}{256}\ast \frac{35}{128}$	$\frac{35}{256}\ast \frac{5}{32}$	$\frac{35}{256}\ast \frac{9}{64}$	$\frac{35}{256}\ast \frac{5}{32}$	$\frac{35}{256}\ast \frac{35}{128}$
2	$\frac{15}{128}\ast \frac{35}{128}$	$\frac{15}{128}\ast \frac{5}{32}$	$\frac{15}{128}\ast \frac{9}{64}$	$\frac{15}{128}\ast \frac{5}{32}$	$\frac{15}{128}\ast \frac{35}{128}$
3	$\frac{15}{128}\ast \frac{35}{128}$	$\frac{15}{128}\ast \frac{5}{32}$	$\frac{15}{128}\ast \frac{9}{64}$	$\frac{15}{128}\ast \frac{5}{32}$	$\frac{15}{128}\ast \frac{35}{128}$
4	$\frac{35}{256}\ast \frac{35}{128}$	$\frac{35}{256}\ast \frac{5}{32}$	$\frac{35}{256}\ast \frac{9}{64}$	$\frac{35}{256}\ast \frac{5}{32}$	$\frac{35}{256}\ast \frac{35}{128}$
5	$\frac{63}{256}\ast \frac{35}{128}$	$\frac{63}{256}\ast \frac{5}{32}$	$\frac{63}{256}\ast \frac{9}{64}$	$\frac{63}{256}\ast \frac{5}{32}$	$\frac{63}{256}\ast \frac{35}{128}$

Normalized form of above table

the below comes from above table

$\ast 2^{15}=32768$

	$s_1=0$	1	2	3	4
$s_2=0$	$2205$	$1260$	$1134$	$1260$	$2205$
1	$1225$	$700$	$630$	$700$	$1225$
2	$1050$	$600$	$540$	$600$	$1050$
3	$1050$	$600$	$540$	$600$	$1050$
4	$1225$	$700$	$630$	$700$	$1225$
5	$2205$	$1260$	$1134$	$1260$	$2205$

Simulation Output

gcc -o Seperate_experiment  Seperate_experiment.c  GSLfun.c -lgsl -lgslcblas -lm
./Seperate_experiment

total sample points: 32768000
Using default random seed
0 2203936 1223708 1050776 1051424 1224938 2206024
1 1257968 701588 599610 601165 699895 1259920
2 1136410 630719 540751 539500 630802 1133816
3 1259445 699621 600675 600773 699740 1259132
4 2204491 1223057 1050313 1050340 1224293 2203170

Combined trials

Possible outcomes of 9 trials

s	0	1	2	3	4	5	6	7	8	9
$C_s^{s-0.5}$	1	$\frac{1}{2}$	$\frac{3}{8}$	$\frac{5}{16}$	$\frac{35}{128}$	$\frac{63}{256}$	$\frac{231}{1024}$	$\frac{429}{2048}$	$\frac{6435}{32768}$	$\frac{12155}{65536}$
$P[s]$	$\frac{12155}{65536}$	$\frac{6435}{65536}$	$\frac{19305}{262144}$	$\frac{2145}{32768}$	$\frac{2205}{32768}$	$\frac{2205}{32768}$	$\frac{2145}{32768}$	$\frac{19305}{262144}$	$\frac{6435}{65536}$	$\frac{12155}{65536}$
$262144\ast P[s]$	$48620$	$25740$	$19305$	$17160$	$17640$	$17640$	$17160$	$19305$	$25740$	$48620$

Combination table

Combination table are calculated using python code

Python Code

# combinations.py
from math import factorial

def nCk(n,k): 
  return factorial(n) / (factorial(k) * factorial(n - k))

S1_element = 4
S2_element = 5
max_number = 3

tuple_list = list()
combination_list = list()
for i in range(max(max_number-S2_element, 0),min(S1_element+1, max_number + 1)):
    tuple_list.append((i, max_number - i))
print(tuple_list)

for item in tuple_list:
    combination_list.append((int(nCk(S1_element, item[0])), int(nCk(S2_element, item[1]))))
print(combination_list)

ratio_list = list()
for item in combination_list:
   ratio_list.append(item[0] * item[1])
ratio_sum = sum(ratio_list)
print(ratio_list)
print(ratio_sum)

s	combinations( $s_1,s_2$)	ratio	probability
0	$(0,0)$	$1$
1	$(0,1),(1,0)$	$5:4$	$\frac{5}{9},\frac{4}{9}$
2	$(0,2),(1,1),(2,0)$	$10:20:6$	$\frac{5}{18},\frac{5}{9},\frac{1}{6}$
3	$(0,3),(1,2),(2,1),(3,0)$	$10:40:30:4$	$\frac{5}{42},\frac{10}{21},\frac{15}{42},\frac{1}{21}$
4	$(0,4),(1,3),(2,2),(3,1),(4,0)$	$5:40:60:20:1$	$\frac{5}{126},\frac{20}{63},\frac{10}{21},\frac{10}{63},\frac{1}{126}$
5	$(0,5),(1,4),(2,3),(3,2),(4,1)$	$1:20:60:40:5$	$\frac{1}{126},\frac{10}{63},\frac{10}{21},\frac{20}{63},\frac{5}{126}$
6	$(1,5),(2,4),(3,3),(4,2)$	$4:30:40:10$	$\frac{1}{21},\frac{15}{42},\frac{10}{21},\frac{5}{42}$
7	$(2,5),(3,4),(4,3)$	$6:20:10$	$\frac{1}{6},\frac{5}{9},\frac{5}{18}$
8	$(3,5),(4,4)$	$4:5$	$\frac{4}{9},\frac{5}{9}$
9	$(4,5)$	$1$

Probability Table

The following table was generated using the code below, some part was from modified version of code above.

Code

# probability.py
from math import factorial

def nCk(n,k): 
  return factorial(n) / (factorial(k) * factorial(n - k))

S1_element = 4
S2_element = 5
max_length = S1_element + S2_element

tuple_matrix = list()
ratio_matrix = list()
sum_list = list()
for current_index in range(max_length + 1):
  tuple_list = list()
  combination_list = list()
  for i in range(max(current_index-S2_element, 0),min(S1_element+1, current_index + 1)):
      tuple_list.append((i, current_index - i))
  print(tuple_list)

  for item in tuple_list:
      combination_list.append((int(nCk(S1_element, item[0])), int(nCk(S2_element, item[1]))))
  print(combination_list)

  ratio_list = list()
  for item in combination_list:
    ratio_list.append(item[0] * item[1])
  ratio_sum = sum(ratio_list)
  print(ratio_list)
  print(ratio_sum)

  tuple_matrix.append(tuple_list)
  ratio_matrix.append(ratio_list)
  sum_list.append(ratio_sum)

# I am too lazy to write a code to generate list of probability
import numpy as np
# P_list = [6435, 3432, 2772, 2520, 2450, 2520, 2772, 3432, 6435]
P_list = [48620, 25740, 19305, 17160, 17640, 17640, 17160, 19305, 25740, 48620]

P_matrix = np.zeros((S1_element + 1, S2_element + 1))

for item_list in tuple_matrix:
    for item, index in zip(item_list, range(len(item_list))):
        first_index = item[0]
        second_index = item[1]
        print(first_index, second_index)

        probability = P_list[first_index + second_index]
        ratio = ratio_matrix[first_index + second_index][index]
        denominator = sum_list[first_index + second_index]
        P_matrix[first_index][second_index] = probability * ratio / denominator

# print(P_matrix) 
for item_list in P_matrix:
    print("|", end="")
    for item in item_list:
        print(item, "|", end="")
    print()

	$s_2=0$	1	2	3	4	5
$s_1=0$	48620.0	14300.0	5362.5	2042.86	700.0	140.0
1	11440.0	10725.0	8171.43	5600.0	2800.0	817.14
2	3217.5	6128.57	8400.0	8400.0	6128.57	3217.5
3	817.14	2800.0	5600.0	8171.43	10725.0	11440.0
4	140.0	700.0	2042.86	5362.5	14300.0	48620.0

Simulation Output

gcc -o Combined_experiment  Combined_experiment.c  GSLfun.c -lgsl -lgslcblas -lm
./Combined_experiment

This could take a while…

total sample points: 262144000
Using default random seed
0 48622130 14298062 5716439 2200221 698170 139973
1 11438522 11441474 8802979 5596545 2799594 880475
2 3430081 6598164 8400003 8403540 6604028 3428588
3 879872 2800758 5600501 8800023 11443106 11438750
4 140201 701729 2203215 5719598 14292911 48624348

tags: 1112_courses probability models