Using Theorem 3.2 and equation (3.13) give another proof for the Cramr-Energy equality on the real line

Using Theorem 3.2 and equation (3.13) give another proof for the Cramér-Energy equality on the real line (Theorem 3.1).

If \(d=1\) then \(c_{d}=c_{1}=\pi\) and thus if we compare the left-hand sides of (3.13) and Theorem 3.1 we get

\[
2 \int_{-\infty}^{\infty}(F(x)-G(x))^{2} d x=\mathcal{E}(X, Y)
\]
which is exactly Theorem 3.1.

Transcribed Image Text:

\(t) - (t)\ dt. 2+ * (F(x) - G(x)} dx = [F(0) = 9(0)1 t