Lecture 2: Story Proofs, Axioms of Probability

Tips for homework

• Labling people, object, etc. -> 1, 2, ... n

  : 10 people, split into team of 6, team of 4 => $\begin{pmatrix}10\\4\\ \end{pmatrix}$ = $\begin{pmatrix}10\\6\\ \end{pmatrix}$
  : 2 teams of 5 => $\begin{pmatrix}10\\5\\ \end{pmatrix}$ //2

• Pick k times from set of n objects, where order doesn't matter, $\begin{pmatrix}n+k-1\\k\\ \end{pmatrix}$ ways.

  • Extreme cases

    : k = 0 => $\begin{pmatrix}n-1\\0\\ \end{pmatrix}$ = 1 not 0

    : k = 1 => $\begin{pmatrix}n\\1\\ \end{pmatrix}$ = n

  • simplest non trivial example

    : n = 2 => $\begin{pmatrix}k+1\\k\\ \end{pmatrix}$ = $\begin{pmatrix}k+1\\1\\ \end{pmatrix}$ = k+1 -> if k = 7) 8

    ex. 2 boxes : | o o o | | o o o o | (# of dots is in {0(not in), 1, ..., 7(all in)})

• Equiv : how many ways are there to put k indistinguishable particles into n distinguishable boxes?

  • n = 4, k = 6 : | o o o | | - | | o o | | o |

    -> | o o o | | o o | o |
    -> k o's, n-1 |'s : k개의 점 사이에 n-1개의 | 를 넣는 방법!

    => $\begin{pmatrix}n+k-1\\k\\ \end{pmatrix}$ = $\begin{pmatrix}n+k-1\\n-1\\ \end{pmatrix}$

  • In physics, this could not be distinguishable. Only for God!
    cf. Bose-Einstein condensate

Story Proof by interpretation

• ex.

  1. $\begin{pmatrix}n\\k\\ \end{pmatrix}$ = $\begin{pmatrix}n\\n-k\\ \end{pmatrix}$

  2. $n\begin{pmatrix}n-1\\k-1\\ \end{pmatrix}$ = $k\begin{pmatrix}n\\k\\ \end{pmatrix}$ : Pick k people out of n, with 1 designated as President

  3. $\begin{pmatrix}m+n\\k\\ \end{pmatrix}$ = $\displaystyle\sum^{k}_{j=0} \begin{pmatrix}m\\j\\ \end{pmatrix} \begin{pmatrix}n\\k-j\\ \end{pmatrix}$ [Vandermonde]

     ex. | o o o | | o o o o o | : pick j in m, pick k-j of n

Non-naive definition

• A probability sample consists of S and P, where S is a sample space, and P, a function which takes an event $A \subseteq S$ as input, returns $P(A) \in [0, 1]$ as output

  • Axioms (공리)

    1. $P(ϕ)=0 (impossible) , P(S)=1$
    2. $P(\displaystyle⋃^{∞}_{n=1} An)=\displaystyle\sum^{k}_{j=0}P(An)$ if A1, A2, ... are disjoint (non-overlap) 서로소

• 출처 : Statistics 110, boostcourse