Exercise 5.4.1. Let Z~Nor (0, 1) and define X₁ = Z and X₂ = Z². Computer₂(X. X.) (XX₂) andrs (X₁, X₂).
Let Z~Normal(0,1). Define X1 = Z and X2 = Z^2
. Calculate the Spearman's coefficient of concordance for the pair (X1, X2).
Hint 1: For Pearson's
symmetry of the normal distribution, or equivalently in terms of calculus, the properties of
properties of the integrals for odd functions. In this way, it will not be necessary to
solve integrals that have no solution in the form of an easy-to-work function.
Hint 2: The way I found to calculate both Kendall's and Spearman's coefficients was to compute the
Spearman coefficients was to calculate the probability P[(X1 - X1
′
)(X2 - X2
′
) > 0] and substituting it into the
forms equivalent to the definition that come in the book. The one that seems simplest to me
is to substitute the X's by their corresponding Z's and use remarkable products to arrive at a
probability in terms of a normal distribution.
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