(Solution) - Consider the Cauchy family defined in Section 3 3 This family -(2025 Original AI-Free Solution)

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Consider the Cauchy family defined in Section 3.3. This family can be extended to a location-scale family yielding pdfs of the form

The mean and variance do not exist for the Cauchy distribution. So the parameters ? and ?2 are not the mean and variance. But they do have important meaning. Show that if X is a random variable with a Cauchy distribution with parameters ? and ?, then:
(a) ? is the median of the distribution of X, that is, P(X > p) = P(X < p) = 1/2.
(b) ? + ? and ? - ? are the quartiles of the distribution of X, that is, P(X > ? + ?) = P(X < ? - ?) = 1/4. (Prove this first for ? = 0 and ? = 1 and then use Exercise 3.38.)