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1 penny / 1 die (Posted on 2011-05-06) Difficulty: 3 of 5
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?

  Submitted by Jer    
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Solution: (Hide)
P(x=12) = dies rolls 6, then 6 heads = 1/6*1/64=1/384
P(y=12) = coin is heads, then two 6s on the dice = 1/2*1/6*1/6=1/72

Since they have a different probability of getting 12, they are not the same distribution.

The mean x is the mean value of the die (3.5) times the mean value of the coin (1.5) which is 5.25
The mean of y is the same thing.

The standard deviation of each can be shown to be √(119)/4

See the first post by Charlie to see pretty good histograms of the distributions.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: standard deviations -- of course --Jer2011-05-08 00:18:22
standard deviations -- of course -- "obvious" problemCharlie2011-05-07 03:28:06
re(2): some solutions, but not sure about standard deviationsCharlie2011-05-06 17:33:19
re: some solutions, but not sure about standard deviationsJer2011-05-06 15:52:41
Solutionsome solutions, but not sure about standard deviationsCharlie2011-05-06 14:12:53
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