An Introduction to Probability Theory and Its Applications, by William Feller

By William Feller

Significant adjustments during this version contain the substitution of probabilistic arguments for combinatorial artifices, and the addition of latest sections on branching tactics, Markov chains, and the De Moivre-Laplace theorem.

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Additional info for An Introduction to Probability Theory and Its Applications, Volume 2

Example text

If one arranges a blind date with a club member, what is the probability that: i) The girl is blonde? ii) The girl is blonde, if it was only revealed that she has blue eyes? 26. 52, what is the probability that: i) The second twin is a boy, given that the first is a boy? ii) The second twin is a girl, given that the first is a girl? 10 Three machines I, II and III manufacture 30%, 30% and 40%, respectively, of the total output of certain items. Of them, 4%, 3% and 2%, respectively, are defective.

Ii) If A ∈ F, then A ∈ Fj for every j ∈ I. Thus Ac ∈ Fj for every j ∈ I, so that Ac ∈ F. iii) If A1, A2 ∈ F, then A1, A2 ∈ Fj for every j ∈ I. Then A1 ∪ A2 ∈ Fj for every j ∈ I, and hence A1 ∪ A2 ∈ F. ▲ THEOREM 2 Let C be an arbitrary class of subsets of S. Then there is a unique minimal field F containing C. ) Clearly, C is contained in the discrete field. Next, let {Fj, j ∈ I} be the class of all fields containing C and define F(C) by PROOF ( ) F C = I F j. j ∈I 10 1 Basic Concepts of Set Theory By Theorem 1, F(C) is a field containing C.

The pair (‫ ޒ‬, B) is called the Borel real line. THEOREM 5 Each one of the following classes generates the Borel σ-field. {(x, y]; x, y ∈‫ޒ‬, x < y}, = {[ x, y); x, y ∈ ‫ ޒ‬, x < y}, = {[ x, y]; x, y ∈ ‫ ޒ‬, x < y}, = {( x, y); x, y ∈ ‫ ޒ‬, x < y}, = {( x, ∞); x ∈ ‫}ޒ‬, = {[ x, ∞); x ∈ ‫}ޒ‬, = {( −∞, x ); x ∈ ‫}ޒ‬, = {( −∞, x ]; x ∈ ‫}ޒ‬. C1 = C2 C3 C4 C5 C6 C7 C8 Also the classes C ′j, j = 1, . . , 8 generate the Borel σ-field, where for j = 1, . . , 8, C′j is defined the same way as Cj is except that x, y are restricted to the rational numbers.

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