=+6.24. Let X be a random variable. Show that for any R, the following two

=+6.24. Let X be a random variable. Show that for any μ ∈ R, the following two conditions (i) and (ii) are equivalent:

(i) nP(|X − μ| > n) → 0 and E{(X − μ)1(|X−μ|≤n)} → 0;

(ii) nP(|X| > n) → 0 and E{X1(|X|≤n)} → μ.

(iii) Use the equivalence of (i) and (ii) to show the necessary and sufficient condition for WLLN given at the end of Section 6.5.