For a Gaussian Bivariate Distribution, the distribution of correlation Coefficients is given by

(1) |

where is the population correlation Coefficient,
is a Hypergeometric Function, and
is the Gamma Function (Kenney and Keeping 1951, pp. 217-221). The Moments are

(2) | |||

(3) | |||

(4) |

where . If the variates are uncorrelated, then and

(5) |

so

(6) |

But from the Legendre Duplication Formula,

(7) |

(8) |

The uncorrelated case can be derived more simply by letting be the true slope, so that
. Then

(9) |

(10) |

(11) |

(12) |

gives

(13) |

so

(14) |

(15) |

Let . For Even , the exponent is an Integer so, by the Binomial Theorem,

(16) |

(17) |

For Odd , the integral is

(18) |

Let so , then

(19) |

But is Odd, so is Even. Therefore

(20) |

Combining with the result from the Cosine Integral gives

(21) |

(22) |

(23) |

(24) |

If , a skew distribution is obtained, but the variable defined by

(25) |

(26) | |||

(27) |

(Kenney and Keeping 1962, p. 266).

Let be the slope of a best-fit line, then the multiple correlation Coefficient is

(28) |

On the surface of a Sphere,

(29) |

(30) | |||

(31) |

Then

(32) |

where

(33) |

**References**

Bevington, P. R. *Data Reduction and Error Analysis for the Physical Sciences.* New York: McGraw-Hill, 1969.

Eckhardt, D. H. ``Correlations Between Global Features of Terrestrial Fields.'' *Math. Geology* **16**,
155-171, 1984.

Kenney, J. F. and Keeping, E. S. *Mathematics of Statistics, Pt. 1, 3rd ed.* Princeton, NJ: Van Nostrand, 1962.

Kenney, J. F. and Keeping, E. S. *Mathematics of Statistics, Pt. 2, 2nd ed.* Princeton, NJ: Van Nostrand, 1951.

Pugh, E. M. and Winslow, G. H. *The Analysis of Physical Measurements.* Reading, MA: Addison-Wesley, 1966.

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1999-05-25