| 15. (a) Smaller values of r give smaller dimensions; larger values of r give larger dimensions. Then the
minimum dimension occurs when all values of r are |
| dim = Log(2)/Log(1/(1/5)) = Log(2)/Log(5) ≈ 0.431. |
| The maximum dimension occurs when all values of r are |
| dim = Log(2)/Log(1/(1/4)) = Log(2)/Log(4) = 0.5. |
| (b) The Moran equation becomes |
| 1 = .25d + .2d. |
| This must be solved numerically, giving d ≈ 0.464, very close, but not equal, to the average of 0.5 and 0.431. |
| (c) With these probabilities, the Moran equation becomes |
| 1 = (4/3)0.25d + (2/3)0.2d |
| Solving numerically gives d ≈ 0.475. Increasing the probability of the larger value of r increases the expected value of the dimension, a sensible result. |
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