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For example, in later chapters we will be much concerned with dice games where the set of interest will contain the possible numbers which can appear on the faces of a dice. We will denote this set by S1 ; then we have S1 = {1, 2, 3, 4, 5, 6}. It is useful to have a notation to indicate when a particular element is a member of a certain set. We use the Greek letter ∈ for this purpose; for example, it is clear that 2 is a member of the set S1 . We write this as 2 ∈ S1 and say that 2 belongs to S1 .

En } be a partition of S with each P (Ei ) = 0 and let A ∈ P (S), then n P (A) = PEi (A)P (Ei ). i=1 Proof You should not find it difficult to verify that the sets A ∩ E1 , A ∩ E2 , . . , A ∩ En yield a partition of A, so that A = (A ∩ E1 ) ∪ (A ∩ E2 ) ∪ · · · ∪ (A ∩ En ). 1). i=1 Application (a simple communication system) Here we begin to make contact with the second main theme of this book, namely the communication of information across a noisy channel which distorts the incoming signal. We will consider the simplest possible model, called the binary symmetric channel, which is sketched in Fig.

We write this as 2 ∈ S1 and say that 2 belongs to S1 . Sometimes we may want to point out that a symbol of interest is not a member of some set. This is carried out using the symbol ∈, so that we have, for example. 7 ∈ S1 . The only example of a set that we have considered so far consists of numbers, but this is by no means typical. For example, the set S2 consists of the names of the planets in the solar system: S2 = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto} . 1 The concept of a set 23 Both of the sets we have considered so far are finite, that is they have only finitely many members; however, many important sets are infinite.

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An introduction to statistical finance by Bouchaud J.-P.


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