A fair six sided die can roll any number from 1 to 6 with equal likelihood.
On fair coin, consider heads to have value 2 and tails to have value 1.

Consider the two experiments:

Experiment A: First roll the die. The outcome tells you how many times to flip the coin. x=the total value of the coin tosses.

Experiment B: First flip the coin. The outcome tells you how many times to roll the die. y=the total value of the die rolls.

1. Prove that the probability distributions of x and y are not the same.
2. How do the means of x and y compare?
3. How do the standard deviations of x and y compare?

    5    dim Prob(12)
   10   for Die=1 to 6
   20    PDie=1/6
   30    for Tot=1 to 12
   40      if Tot>=Die and Tot<=2*Die then
   50         :PTot=PDie*combi(Die,Tot-Die)/2^Die
   60         :Prob(Tot)=Prob(Tot)+PTot
   70    next Tot
   80   next Die
   89   Sum=0
   90   for I=1 to 12
  100      print I,Prob(I);tab(22);Prob(I)/1;tab(45);:Cum=Cum+Prob(I)
  101      for J=1 to int(150*Prob(I)):print "*";:next:print
  102      Sum=Sum+I*Prob(I)
  110   next
  120   print Cum,Sum,Sum/1,
  130   for I=1 to 12
  140      Sumsq=(I-Sum)*(I-Sum)*Prob(I)
  150   next I
  160   Sd=sqrt(Sumsq)
  170   print Sd
  200   print
  210   for I=1 to 12:Prob(I)=0:next I
  310   for Coin=1 to 2
  320    PCoin=1/2
  330    for Die=1 to 6
  335      if Coin=1 then
  340       :Prob(Die)=Prob(Die)+PCoin/6
  345      :else
  350        :for Die2=1 to 6
  355          :Prob(Die+Die2)=Prob(Die+Die2)+PCoin/36
  360        :next Die2
  365      :endif
  370    next Die
  380   next Coin
  389   Sum=0:Cum=0:Sumsq=0
  390   for I=1 to 12
  400      print I,Prob(I);tab(22);Prob(I)/1;tab(45);:Cum=Cum+Prob(I)
  401      for J=1 to int(150*Prob(I)):print "*";:next:print
  402      Sum=Sum+I*Prob(I)
  410   next
  420   print Cum,Sum,Sum/1,
  430   for I=1 to 12
  440      Sumsq=(I-Sum)*(I-Sum)*Prob(I)
  450   next I
  460   Sd=sqrt(Sumsq)
  470   print Sd

finds

Experiment A:

total            probability                text graph
 1       1/12         0.0833333333333333333 ************
 2       1/8          0.125                 ******************
 3       5/48         0.1041666666666666666 ***************
 4       11/96        0.1145833333333333333 *****************
 5       7/64         0.109375              ****************
 6       43/384       0.1119791666666666666 ****************
 7       7/64         0.109375              ****************
 8       13/128       0.1015625             ***************
 9       5/64         0.078125              ***********
 10      17/384       0.0442708333333333333 ******
 11      1/64         0.015625              **
 12      1/384        0.0026041666666666666
 
            mean            s.d.
 1       21/4   5.25    0.34445949507888442
 

Experiment B:

 1       1/12         0.0833333333333333333 ************
 2       7/72         0.0972222222222222221 **************
 3       1/9          0.111111111111111111  ****************
 4       1/8          0.125                 ******************
 5       5/36         0.1388888888888888888 ********************
 6       11/72        0.1527777777777777777 **********************
 7       1/12         0.0833333333333333333 ************
 8       5/72         0.0694444444444444444 **********
 9       1/18         0.0555555555555555555 ********
 10      1/24         0.0416666666666666666 ******
 11      1/36         0.0277777777777777777 ****
 12      1/72         0.0138888888888888888 **
 
            mean            s.d.
 1       21/4   5.25    0.7954951288348659649
 

The means agree with what would be expected, as 1.5 * 3.5, which is the expected value of the coin tosses times the expected value on the die.

However, the standard deviation seems larger in experiment B while the data would seem to indicate it should be the other way around.

The calculations are in the program shown above.

 

Posted by Charlie on 2011-05-06 14:12:53