Notes - Probability MT22, Chapter 2
What are the two conditions for function $p _ X(x)$ to really be a probability mass function?
- $\forall x: p _ X(x) \ge 0$
- $\sum _ {x\in\text{Im }X} p _ X(x) = 1$
Given a function $p _ X(x)$ that satisfies the conditions for being a probability mass function, how could we define a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a random variable $X$ that corresponds to this probability mass function?
- $\Omega = \{x \in \mathbb{R} : p(x) \ne 0\}$
- $\mathcal{F} = \mathcal{P}(\Omega)$
- $\mathbb{P}(S) = \sum _ {\omega \in S} p(\omega)$
- $X(\omega) = \omega$
Suppose B is an event such that $\mathbb{P}(B) \ne 0$. What is the definition of the “conditional mass function of X given B”?
\[\mathbb{P}(X = x|B) = \frac{\mathbb{P}(\\{X = x\\} \cap B)}{\mathbb{P}(B)}\]
What is $\mathbb{E}[h(X, Y)]$ for discrete random variables $X$ and $Y$?
\[\sum_{x \in \text{Im }X} \sum_{y \in \text{Im }Y} h(x, y) \mathbb{P}(X = x, Y = y)\]