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@@ V následující větě shrneme pár základních vlastností prostoru $\mathcal Dále můžeme počítat $$ \mathbb{E}\left[\prod_{d=1}^l X_l \right] = \int_{\mathbb{R}^d} \prod_{l=1}^d x_l dP_{\vec{X}} = \int_{\mathbb{R}} \prod_{l=1}^d x_l d(\otimes_{l=1}^d P_{X_l}) = $$ $$ \int_{\mathbb{R}}\cdots\int_{\mathbb{R}} \prod_{l=1}^d x_l dP_{X_1} \cdots dP_{X_d} = \prod_{l=1}^d \int_\mathbb{R} x_l dP_{X_l} = \prod_{l=1}^d \mathbb{E}X_l, $$ - kde druhá rovnost plyne z nezávislosti, třetí z Fubiniovy věty a předposlední z linearity integrálu. + kde druhá rovnost plyne z nezávislosti náhodných veličin $X_1,\dots,X_d$, třetí z Fubiniovy věty a předposlední z linearity integrálu. \end{proof} \hfill \textit{konec 8. přednášky (11.3.2025)}