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Last updated Nov 1, 2022
...\begin{align*} &\mathbb{E}(e^{tX} 1_{X \geq 0}) \leq \mathbb{E}(e^{tX})\\ &\mathbb{E}(e^{-tX} 1_{X < 0}) \leq \mathbb{E}(e^{-tX})\\ &\mathbb{E}(e^{tX} 1{X \geq 0}) + \mathbb{E}(e^{-tX} 1{X < 0}) = \mathbb{E}(e^{t|X|}) \leq m{X}(t) + m{X}(-t) \end{align*} Other Outlinks Encounters - Ch 9......
11/7/2022
...F_{X}(dx)\\ &=\int\limits\mathbb{R} |x+y-y| F{X}(dx)\\ &\leq \int\limits\mathbb{R} |x+y| F{X}(dx) + \int\limits\mathbb{R} |y| F{X}(dx)\\ &=\int\limits\mathbb{R} |x+y| F{X}(dx) + |y|\\ &< \infty \end{align*}$$ and $X \in L^{1}(\Omega)$. Swap $X$ and $Y$ to get the same result for $Y$. $\checkmark$ $\blacksquare$ Other Outlinks ......
11/7/2022