| |
\[
\begin{aligned}
\mathbb{E}[X] &= \sum_x x\cdot p_X(x) \qquad\text{or} \quad \int x\cdot f_X(x)\,dx \\
\mathbb{E}[X\,|\,A] &= \sum_x x\cdot p_{X|A}(x) \qquad\text{or} \quad \int x\cdot f_{X|A}(x)\,dx \\
g(y) &= \mathbb{E}[X\,|\,Y=y] = \sum_x x\cdot p_{X|Y}(x\,|\,y) \qquad\text{or} \quad \int x\cdot f_{X|Y}(x\,|\,y)\,dx \\
g(Y) &= \mathbb{E}[X\,|\,Y] & \text{conditional expectation as r.v.} \\
\end{aligned}
\] |
Expected value rule
| |
\[
\begin{aligned}
\mathbb{E}[g(X)] &= \sum_x g(x)\cdot p_X(x) \qquad\text{or} \quad \int g(x)\cdot f_X(x)\,dx \\
\mathbb{E}[g(X)\,|\,A] &= \sum_x g(x)\cdot p_{X|A}(x) \qquad\text{or} \quad \int g(x)\cdot f_{X|A}(x)\,dx \\
\mathbb{E}[g(X,Y)] &= \sum_x\sum_y g(x,y)\cdot p_{X,Y}(x,y) \qquad\text{or} \quad \int\int g(x,y)\cdot f_{X,Y}(x,y)\,dx\,dy \\
\end{aligned}
\] |
Total probability and expectation theorems
| |
\[
\begin{aligned}
\mathbb{P}(B) &= \color{blue}{\mathbb{P}(A_1)}\cdot \mathbb{P}(B\,|\,A_1) + \cdots + \color{blue}{\mathbb{P}(A_n)}\cdot \mathbb{P}(B\,|\,A_n) \\
p_X(x) &= \color{blue}{\mathbb{P}(A_1)}\cdot p_{X|A_1}(x) + \cdots + \color{blue}{\mathbb{P}(A_n)}\cdot p_{X|A_n}(x) \\
\\
F_X(x) &= \mathbb{P}(X \le x) \\
&= \color{blue}{\mathbb{P}(A_1)}\cdot \mathbb{P}(X \le\,|\,A_1) + \cdots + \color{blue}{\mathbb{P}(A_n)}\cdot \mathbb{P}(X \le x\,|\,A_n) \\
&= \color{blue}{\mathbb{P}(A_1)}\cdot F_{X|A_1}(x) + \cdots + \color{blue}{\mathbb{P}(A_n)}\cdot F_{X|A_n}(x) \\
\\
f_X(x) &= \color{blue}{\mathbb{P}(A_1)}\cdot f_{X|A_1}(x) + \cdots + \color{blue}{\mathbb{P}(A_n)}\cdot f_{X|A_n}(x) \\
\int x\cdot f_X(x)\,dx &= \color{blue}{\mathbb{P}(A_1)}\int x\cdot f_{X|A_1}(x)\,dx + \cdots + \color{blue}{\mathbb{P}(A_n)}\int x\cdot f_{X|A_n}(x)\,dx \\
\mathbb{E}[X] &= \color{blue}{\mathbb{P}(A_1)}\cdot \mathbb{E}[X\,|\,A_1] + \cdots + \color{blue}{\mathbb{P}(A_n)}\cdot \mathbb{E}[X\,|\,A_n] \\
\mathbb{E}[X_1 + \cdots + X_N] &= \sum_n p_N(n)\cdot \mathbb{E}\big[X_1 + \cdots + X_N \,\big|\, N=n \big] \\
& = \color{red}{\sum_n p_N(n)\cdot n \cdot \mathbb{E}[X]} = \mathbb{E}[N]\cdot\mathbb{E}[X] \\
\end{aligned}
\] |
Linearity of expectations
| |
\[
\begin{aligned}
\mathbb{E}[aX+b] &= a\mathbb{E}[X] + b \\
\mathbb{E}[X + Y] &= \mathbb{E}[X] + \mathbb{E}[Y] \\
\end{aligned}
\] |
Law of iterated expectations
| |
\[
\begin{aligned}
\mathbb{E}\left[\mathbb{E}[X\,|\,Y]\right] &= \sum_y \mathbb{E}[X\,|\,Y=y] \cdot p_Y(y) = \mathbb{E}[X] \\
\mathbb{E}\left[\mathbb{E}[X_1 + \cdots + X_N\,|\,N]\right] &= \color{red}{\mathbb{E}\big[N\cdot\mathbb{E}[X]\big]} = \mathbb{E}[N]\cdot\mathbb{E}[X] \\
\end{aligned}
\] |
Source: MITx 6.041x, Lecture 9, 13.
# posted by rot13(Unafba Pune) @ 9:10 AM
