lecture_04

Wed Jun 13 09:02:27 CDT 2012

• recall

  • [ (x + y)^n = \sum\limit_{k=0}^n { \frac{ n!}{ k!(n - k)!} x^k y^{n-k} ]
    • [ = \sum\limits_{ a + b - n , a \ge 0, b \ge 0}{ \frac{ n!}{ a! b!} x^a y^b ]
  • therefore
    • [ (x_1 + x_2)^n = \sum\limits_{ n_1 + n_2 = n,n_1 \ge 0, n_2 \ge 0}{ \frac{ n!}{ n_1! n_2!} x_1^{n_1} x_2^{n_2} ]

• Trinomial Theorem

  • [ (x_1 + x_2 + x_3)^n = \sum\limits_{ n_1 + n_2 + n_3 = n, n_i \ge 0}{ \frac{ n!}{ n_1! n_2! n_3!} x_1^{n_1} x_2^{n_2} x_3^{n_3} } ]

• Multinomial Theorem

  • [ \left( \sum\limits_{ i = 1}^I{ x_i} \right)^n = \sum\limits_{ n_1 + \cdots n_I = n, n_i \ge 0}{ \frac{ n!}{ n_1! n_2! \cdots n_I!} x_1^{n_1} \cdots x_I^{n_I} ]

Axromatic Probability

• [ { A \cup B}: \forall x \in S s.t. x \in A or x \in B ]

Facts from set Theory

[ A \cap \left( B \cup C \right) = \left( A \cap B \right) \cup \left( A \cap B \right)] [ A \left( B \cup C \right) = AB \cup AC] [ A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left( A \cup C \right)]

[ left( \bigcup\limits_{ i\in I}{ A_i} \right) = \left( \bigcap\limits_{ i \in I}{ A_i^c \right) ]

[ left( A \cup B \right)^c = A^c \cap B^c ]

• A probability theory is a triple
  • [ \left( S, { E_i }, P \right ]
  • satisfying 3 axioms:
    • A.1 [ 0 \le P(E) \le 1 ]
    • A.2 [ P(S) = 1 ]
    • A.3 [ P\left( \bigcup\limits_{ i = 1}^{ \infty}{ E_i} \right) = \sum\limits_{ i = 1}{ \infty}{ P\left( E_i \right) }] whenever [ E_i \cap E_2 = \emtyset \forall i \notequal j]