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- STA256 Lecture 03

Continuity Theorem of Probability

${C_{n}}$ is an increasing sequence of events
$lim_{n \to \infty} P (C_{n}) = P (lim_{n \to \infty} C_{n}) = P (⋂_{n = 1}^{\infty} C_{n})$
For increasing sequences:
- $lim_{n \to \infty} P (C_{n}) = P (lim_{n \to \infty} C_{n}) = P (⋃_{n = 1}^{\infty} C_{n})$
- Proof:
  - Let ${C_{n}}$ be an increasing sequence
  - By definition of an increasing sequence (terms are getting larger)
    - So $C_{1} \subset C_{2} \subset C_{3} \subset \dots \subset C_{n}$
    - ^ That's an increasing sequence
  - Define the mutually exclusive regions
  - $\begin{matrix} R_{1} = C_{1} \\ R_{2} = C_{2} - C_{1} \\ R_{3} = C_{3} - C_{2} \\ \dots \\ R_{n} = C_{n} - C_{n - 1} \end{matrix}$
  - $C_{n}$ is an increasing sequence
    - $lim_{n \to \infty} C_{n} = ⋃_{n = 1}^{\infty} C_{n}$
    - $lim_{n \to \infty} C_{n} = ⋃_{j = 1}^{\infty} R_{j}$
    - $lim_{n \to \infty} C_{n} = R_{1} \cup R_{2} \cup R_{3} \cup \dots R_{j}$
    - $P (lim_{n \to \infty} C_{n}) = P (R_{1} \cup R_{2} \cup R_{3} \cup \dots \cup R_{j})$
      - The regions of $R$ are mutually exclusive.
      - By axiom 3, we can just add the probabilities together
    - $P (lim_{n \to \infty} C_{n}) = P (R_{1}) + P (R_{2}) + P (R_{3}) + \dots + P (R_{j})$
    - $P (lim_{n \to \infty} C_{n}) = P (C_{1}) + P (C_{2}) - P (C_{1}) + P (C_{3}) - P (C_{2}) + \dots + P (C_{n}) - P (C_{n - 1})$
    - $P (lim_{n \to \infty} C_{n}) = lim_{n \to \infty} P (C_{n})$
For decreasing sequences:
- ${A_{n}}$ is decreasing
- $A_{1} \supset A_{2} \supset A_{3} \supset \dots \supset A_{n}$
- ${A_{n}^{c}}$ is increasing
- $A_{1}^{c} \subset A_{2}^{c} \subset A_{3}^{c} \subset \dots \subset A_{n}^{c}$