2.2 The purpose of this exercise is to verify the density in (2.19). Without loss of generality let u = (1,0, ..., 0)

T, so ????Tu = ????[1]. Using Exercise 2.1(c), show that P(????[1] < s) = ????d−1

????d ∫

s

−1

(1 − ????2

)

(d−3)∕2 d????.

Differentiate with respect to s to get the required density. See also Mardia and Jupp (2000, p. 167).