Lecture - Probability MT22, VI
Flashcards
How could you calculate $\mathbb{P}(X = x \vert B)$?
\[\frac{\mathbb{P}(X=x, B)}{\mathbb{P}(B)}\]
What’s the formula for $\mathbb{E}[X \vert B]$?
\[\sum_{\text{Im} X} x\mathbb{P}(X = x |B)\]
The Law of Total Probability or partition theorem states that if $\{B _ i, i \in I\}$ is a countable partition of $\Omega$, then
\[\mathbb{P}(A) = \sum _ {i \in I}\mathbb{P}(A \vert B _ i)\mathbb{P}(B _ i)\]
What’s the partition theorem for expectations?
\[\mathbb{E}[X] = \sum_{i \in I} \mathbb{E}[X | B_i] \mathbb{P}(B_i)\]
What must hold about any joint probability mass function $p _ {X, Y}(x, y)$?
\[\sum_{x\in\text{Im}X} \sum_{y\in\text{Im}Y} p_{X,Y}(x, y) = 1\]
How can you determine the marginal distribution $\mathbb{P}(X = x)$ given the joint distribution $\mathbb{P}(X = x, Y = y)$?
\[\mathbb{P}(X = x) = \sum_{y \in \text{Im} Y} \mathbb{P}(X = x, Y = y)\]
What is true about the joint distribution $p _ {X,Y}(x,y)$ and the marginal distributions $p _ X(x)$ and $p _ Y(y)$ when $X$ and $Y$ are independent?
\[p_{X,Y}(x, y) = p_X(x) p_Y(y)\]
What is $\mathbb{E}[h(X, Y)]$?
\[\sum_{x \in \text{Im}X}\sum_{y \in \text{Im} Y} h(x, y) p_{X,Y}(x, y)\]
How can you rewrite $\mathbb{E}[aX + bY]$?
\[a \mathbb{E}[X] + b\mathbb{E}[Y]\]