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Postscript version of this file

STAT 870 Lecture 1

Course outline

Goals of Today's Lecture:

Course outline:

Course outline continued: Course outline continued:
Models for coin tossing

Coin tossing example: n=3.

Complicated Events: examples


\begin{displaymath}A_1\equiv \{\omega: \lim_{n\to\infty} (\omega_1+\cdots + \omega_n)/n \text{
exists }\}
\end{displaymath}


\begin{displaymath}A_2 \equiv \{\omega: \lim_{n\to\infty} (\omega_1+\cdots + \omega_n)/n
=1/2\}
\end{displaymath}


\begin{displaymath}A_3 \equiv \{\omega: \lim_{n\to\infty} \sum_1^n (2\omega_k-1)/k \text{
exists }\}
\end{displaymath}

Some mathematical questions to answer:

1.
Do (4) and (5) determine P(A) for any $A\subset\Omega$? [NO]

2.
Do (4) and (5) determine P(Ai) for i=1,2,3? [YES]

3.
Are (4) and (5) logically consistent? [YES]

Probability Definitions

Probability Space (or Sample Space): ordered triple $(\Omega, {\cal F}, P)$.

Axioms guarantee can compute probabilities by usual rules, including approximation without contradiction.

Consequences:

1.
Finite additivity if $A_1,A_2,\cdots,A_n$ pairwise disjoint:

\begin{displaymath}P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i) \, .
\end{displaymath}

2.
For any event A P(Ac) = 1-P(A) .

3.
If $A_1 \subset A_2 \subset \cdots$ are events then

\begin{displaymath}P(\bigcup_1^\infty A_i) = \lim_{n\to\infty} P(A_n) \, .
\end{displaymath}

4.
If $A_1 \supset A_2 \supset \cdots$ then

\begin{displaymath}P(\bigcap_1^\infty A_i) = \lim_{n\to\infty} P(A_n) \, .
\end{displaymath}

Most subtle point is $\sigma$-field, $\cal F$. Needed to avoid some contradictions which arise if you try to define P(A) for every subset A of $\Omega$when $\Omega$ is a set with uncountably many elements.

Random Variables:

Vector valued random variable: function X, domain $\Omega$, range in ${\Bbb R}^p$ such that

\begin{displaymath}P(X_1 \le x_1, \ldots , X_p \le x_p)
\end{displaymath}

is defined for any constants $(x_1,\ldots,x_p)$. Notation: $X=(X_1,\ldots,X_p)$ and

\begin{displaymath}X_1 \le x_1, \ldots , X_p \le x_p
\end{displaymath}

is shorthand for an event:

\begin{displaymath}\left\{\omega\in\Omega:
X_1(\omega) \le x_1, \ldots , X_p (\omega) \le x_p \right\}
\end{displaymath}

X function on $\Omega$ so X1 function on $\Omega$.

Formal definitions:

The Borel $\sigma$-field in ${\Bbb R}^p$ is the smallest $\sigma$-field in ${\Bbb R}^p$ containing every open ball

\begin{displaymath}B_y(r) = \{ x\in {\Bbb R}^p: \vert x-y\vert < r\}\, .
\end{displaymath}

(To see that there is, in fact, such a ``smallest'' $\sigma$-field you prove the following assertions:

1.
The intersection of an arbitrary family of $\sigma$-fields is a $\sigma$-field. [Homework set 1.]

2.
There is at least one $\sigma$-field of subsets of ${\Bbb R}^p$ containing every open ball. [Homework set 1.]

Now define the Borel $\sigma$-field in ${\Bbb R}^p$ to be

\begin{displaymath}{\cal B}({\Bbb R}^p) = \cap {\cal F}
\end{displaymath}

where the intersection runs over all $\sigma$-fields $\cal F$which contain every open ball.)


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Richard Lockhart
2000-09-12