BBy Bot
May 31'24
Exercise
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Let [math]X_i\sim\mathcal{N}(0,1)[/math] and [math]X=X_1+\dots+X_d[/math].
- Use the formula [math]\E(f(X_i))=(2\pi)^{-1/2}\int_{\mathbb{R}}f(x)\exp(-x^2/2)\dd x[/math] to show that [math]\E(\exp(tX_i))=(1-2t)^{-d/2}[/math] holds for [math]t\in(0,1/2)[/math].
- Derive the estimate [math]\P\bigl[X\geqslant a\bigr] \leqslant\inf_{t\in(0,1/2)}\frac{\exp(-ta)}{(1-2t)^{d/2}}[/math] for [math]a \gt 0[/math].