1 + M2 = l(M, N), say, and the limit of l(M, N) as M, N →∞depends on the way in which M and N approach

∞. If M → ∞ and N → ∞ in that order, then l(M, N) → −∞, while if the limit is taken in the other order, then l(M, N) → ∞. Hence the Cauchy distribution does not have a mean value. On the other hand, there are many functions of X with finite expectations. For example, if Y = tan−1 X, then E(Y ) =

Z

∞

−∞

tan−1 x 1

π(1 + x2)

dx

=

Z 1 2 π

−1 2 π

v

π

dv where v = tan−1 x

= 0. △