Characterizations of Correlation Coefficient

This is the authors' abstract. We don't add key points for this paper.

The probability density function of correlation coefficient conditional of chi-square variables has been derived. Then the pdf of the correlation coefficient has been easily derived. The conditional moments of correlation coefficient have been derived. The well known moments of correlation coefficient can be easily and quickly derived from the conditional moments. The intermediate steps of the derivation in terms of conditional distributions or moments seem to be worth recording and will be instructive to learners and instructors. A random decomposition of sample correlation coefficient has been discussed. An algorithm has been presented to calculate sample correlation coefficient by using only sample variance.