Show that the differential equation of Exercise 30 with b = c = 0 and a > 0 yields a normal distribution.

Exercise 30

Karl Pearson, one of the founders of modern statistics, showed that the differential equation

yields (for appropriate values of the constants a, b, c, and d) most of the important distributions of statistics. Verify that the differential equation gives

(a) the gamma distribution when a = c = 0, b > 0, and d > -b;

(b) the exponential distribution when a = c = d = 0 and b > 0;