The density of the correlation coefficient r of a sample of size N = n+1 from a bivariate normal distribution with correlation ρ is given by                     f (r | ρ) = (n − 1)(n) √2π (1 − ρ2 )...

The density of the correlation coefficient r of a sample of size N = n+1 from a bivariate normal distribution with correlation ρ is given by

                    f (r | ρ) = (n − 1)(n) √2π (1 − ρ2 ) n/2 (1 − ρr)−n+1/2 (1 − r 2 ) (n−3)/2 ×

                           ∞ j=0  j + 1 2 2  1 2 2  n + j + 1 2  (1 + ρr)j 2j j! .

(a) Simplify the gamma functions to facilitate computation.

(b) Write code to compute this function for any value of r and ρ.

(c) Plot the density for ρ = ±1/3 and 0 and check symmetry.