# Question: Karl Pearson one of the founders of modern statistics showed

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.

## Answer to relevant Questions

Show that the normal distribution has (a) A relative maximum at x = µ; (b) Inflection points at x = µ – σ and x = µ + σ. If X is a random variable having the standard nor–mal distribution and Y = X2, show that cov(X, Y) = 0 even though X and Y are evidently not independent. To prove Theorem 6.10, show that if X and Y have a bivariate normal distribution, then (a) Their independence implies that ρ = 0; (b) ρ = 0 implies that they are independent. Theorem 6.10 If two random variables have a ...In a certain city, the daily consumption of electric power in millions of kilowatt– hours can be treated as a random variable having a gamma distribution with α = 3 and β = 2. If the power plant of this city has a daily ...Suppose that the service life in hours of a semiconductor is a random variable having a Weibull distribution (see Exercise 6.23) with α = 0.025 and β = 0.500. (a) How long can such a semiconductor be expected to last? ...Post your question