# Question

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;

(c) The beta distribution when a = 0, b = – c, d– 1 / b < 1, and db > – 1.

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;

(c) The beta distribution when a = 0, b = – c, d– 1 / b < 1, and db > – 1.

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