----- > [!proposition] Proposition. ([[expectation is multiplicative for independent random variables]]) > Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a [[probability|probability space]] and $X,Y$ are [[independent random variables]] [[Lp-norm|in]] $\mathcal{L}^{2}(\mathbb{P})$. Then $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y].$ ^proposition > [!proof]- Proof. ([[expectation is multiplicative for independent random variables]]) > We will prove the result for $X,Y$ [[random variable|simple random variables]], then take a limit to argue for the general case. > > **Part 1.** [[indicator random variable|So suppose]] $X=a_{1} 1_{\{ X =a_{1} \}}+\dots + a_{M} 1_{\{ X=a_{M} \}}$ and $Y=b_{1} 1_{\{ Y=b_{1} \}}+\dots+ b_{N} 1_{\{ Y=b_{N} \}}.$ Now $\begin{align} > XY= \sum_{j=1}^{M} \sum_{k=1}^{N} a_{j} b_{k} 1_{\{ X=a_{j} \} \cap \{ Y=b_{k} \}}, > \end{align}$ > and thus $\begin{align} > \mathbb{E}[XY]&= \sum_{j=1}^{M} \sum_{k=1}^{N} a_{j} b_{k} \overbrace{ \mathbb{E}[1_{\{ X=a_{j} \} \cap \{ Y=b_{k} \}}] }^{ = \mathbb{P}(\{ X=a_{j} \} \cap \{ Y=b_{k} \}) } \\ > &= \sum_{j=1}^{M} \sum_{k=1}^{N} a_{j}b_{k} \underbrace{ \mathbb{P}(\{ X=a_{j} \})\mathbb{P}(Y=b_{k}) }_{ = \mathbb{E}1_{\{ X=a_{j} \}} \cdot \mathbb{E}1_{\{ Y=b_{k} \}} } \\ > &= \mathbb{E}\left[ \sum_{j=1}^{M} a_{j} 1_{\{ X=a_{j} \}} \right] \mathbb{E}\left[\sum_{k=1}^{N} b_{k} 1_{\{ Y=b_{k} \}}\right] \\ > &= \mathbb{E}[X] \cdot \mathbb{E}[Y], > \end{align}$ > where the second equality uses independence of $X$ and $Y$. This proves the result in the case of two [[random variable|simple random variables]]. > > **Part 2.** Now consider arbitrary independent random variables $X,Y$ in $\mathcal{L}^{2}(\mathbb{P})$. Let $f_{1},f_{2},\dots: \mathbb{R} \to \mathbb{R}$ be a [[sequence]] of [[measurable function|Borel measurable]] [[simple function|simple functions]] the approximate the [[identity map]] on $\mathbb{R}$, [[approximation by simple functions|in the sense that]] $(f_{n}(t))_{n=1}^{\infty} \uparrow t$ for every $t \in \mathbb{R}$. The random variables $f_{n} \circ X$ and $f_{n} \circ Y$ are independent [[independent random variables|(see)]]; thus the result in the paragraph above says $\mathbb{E}[(f_{n} \circ X)(f_{n} \circ Y)] =\mathbb{E}[f_{n} \circ X] \cdot \mathbb{E}[f_{n} \circ Y] \text{ for each }n \in \mathbb{N}.$ > [[pointwise convergence|Noting]] the immediate fact $X=\lim_{n \to \infty}(f_{n} \circ X)$ and similar for $Y$, we have $\begin{align} > \mathbb{E}[XY]&=\mathbb{E}[ \lim_{n \to \infty} \big( (f_{n} \circ X ) \cdot ( f_{n} \circ Y)\big)] \\ > &= \lim_{n \to \infty} \mathbb{E}[(f_{n} \circ X) \cdot (f_{n} \circ Y)]\\ > &= \lim_{n \to \infty} \mathbb{E}[f_{n} \circ X] \cdot \lim_{n \to \infty} \mathbb{E}[f_{n} \circ Y] \\ > &= \mathbb{E}[ \lim_{n \to \infty} (f_{n} \circ X)] \cdot \mathbb{E}[ \lim_{n \to \infty} (f_{n} \circ Y)] \\ > &= \mathbb{E}[X] \mathbb{E}[Y], > \end{align}$ > where the exchanging of [[converge|limits]] and [[expectation|expectations]] has been valid in light of the [[Dominated Convergence Theorem]]. [[Dominated Convergence Theorem|DCT]] is in turn valid due to fact that $|f_{n} \circ X| \leq |X|$ and $|f_{n} \circ Y| \leq |Y|$ for all $n$, plus the fact $|(f_{n} \circ X) (f_{n} \circ Y) | \leq |X| |Y|$. (These dominators are [[integrable]] as required by [[Dominated Convergence Theorem|DCT]] by [[Hölder's inequality]] $\int |X| |Y| \, d\mu \leq \|X\|_{2} \|Y\|_{2} < \infty$ since we assumed $X,Y \in \mathcal{L}^{2}(\mathbb{P})$.) > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```