Prove (20.20), i.e. show that (mathbb{E}left[B_{T}^{2 n} e^{-B_{T}^{2}} ight]=frac{(2 n-1) ! !}{sqrt{2 T+1}}left(frac{T}{2 T+1} ight)^{n}). Here, ((-1)

Prove (20.20), i.e. show that \(\mathbb{E}\left[B_{T}^{2 n} e^{-B_{T}^{2}}\right]=\frac{(2 n-1) ! !}{\sqrt{2 T+1}}\left(\frac{T}{2 T+1}\right)^{n}\). Here, \((-1) ! !:=1\), \((2 n-1) ! !=1 \cdot 3 \cdot \ldots \cdot(2 n-1)\), is the double factorial.

Data From Formula (20.20)

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E[B]= e-B (2n-1)!! n (20.20) 2T+1 2T+1