06-01.md

title	date
Class 2	January 06, 2023

• Random Experiments and Sample Space ($\Omega$)
• Probability Measure ($\mathbb{P}$) and it's axioms
• Sigma-Algebra ($\mathcal{F}$)
  • Null set
  • Element and it's complement
  • Closure under countable union of disjoint element
• Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$
  • When $\Omega = \mathbb{R}$
• Conditional Probability
  • Chain rule $$P(A_1 \cap A_2 \ldots A_n) = P(A_1) P(A_2 | A_1) P(A_3 | A_1 \cap A_2) \ldots P(A_n | A_1 \cap A_2 \cap \ldots A_{n-1})$$

• Independent Events
• Conditional Independence $P(AB | C) = P(A|C) P(B|C)$
• Mutually exclusive and Independence
  • If one has zero-probability then related.
• Random Variable
  • Map from one probability space to another
  • Most cases the resultant probability space has sample space $\mathbb{R}$. $$\therefore \ (\Omega,\ \mathcal{F},\ \mathbb{P}(.)) \ \stackrel{X}{\longrightarrow} \ (\mathbb{R}, \ \mathcal{B}(\mathbb{R}),\ \mathbb{P}_X(.))$$
    • $X^{-1} (B)$ is called the preimage or the inverse image of $B$.
  • Discrete vs Continuous R.V.
  • nth moment $E[X^n]$
  • Law of unconscious statistician $E[g(X)] = \sum g(x) p_X(x)$.
  • Joint Random Variabless