Dadidadida: Expectation

Friday, April 04, 2014

Expectation

$\begin{aligned} \mathbb{E}[X] &= \sum_x x\cdot p_X(x) \qquad\text{or} \quad \int x\cdot f_X(x)\,dx \\ \mathbb{E}[X\,|\,A] &= \sum_x x\cdot p_{X|A}(x) \qquad\text{or} \quad \int x\cdot f_{X|A}(x)\,dx \\ g(y) &= \mathbb{E}[X\,|\,Y=y] = \sum_x x\cdot p_{X|Y}(x\,|\,y) \qquad\text{or} \quad \int x\cdot f_{X|Y}(x\,|\,y)\,dx \\ g(Y) &= \mathbb{E}[X\,|\,Y] & \text{conditional expectation as r.v.} \\ \end{aligned}$

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

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Friday, April 04, 2014

Expectation

Expected value rule

Total probability and expectation theorems

Linearity of expectations

Law of iterated expectations

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