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ISIPTA'09 -
SIXTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Durham University, Department of Mathematical Sciences

Durham, United Kingdom

Tuesday 14 to Saturday 18 July 2009

## ELECTRONIC PROCEEDINGS

## Veronica Biazzo, Angelo Gilio, Giuseppe Sanfilippo

# On General Conditional Random Quantities

### Abstract

In the first part of this paper, recalling a general discussion on iterated conditioning given by de Finetti in the appendix of his book, vol. 2, we give a representation of a conditional random quantity $X|HK$ as $(X|H)|K$. In this way, we obtain the classical formula $\pr{(XH|K)} =\pr{(X|HK)P(H|K)}$, by simply using linearity of prevision. Then, we consider the notion of general conditional prevision $\pr(X|Y)$, where $X$ and $Y$ are two random quantities, introduced in 1990 in a paper by Lad and Dickey. After recalling the case where $Y$ is an event, we consider the case of discrete finite random quantities and we make some critical comments and examples. We give a notion of coherence for such more general conditional prevision assessments; then, we obtain a strong generalized compound prevision theorem. We study the coherence of a general conditional prevision assessment $\pr(X|Y)$ when $Y$ has no negative values and when $Y$ has no positive values. Finally, we give some results on coherence of $\pr(X|Y)$ when $Y$ assumes both positive and negative values. In order to illustrate critical aspects and remarks we examine several examples.

** Keywords. ** conditional events, general conditional random quantities, general conditional prevision assessments, generalized compound prevision theorem, iterated conditioning, strong generalized compound prevision theorem.

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** Authors addresses: **

**Veronica Biazzo**

Dipartimento di Matematica e Informatica

Universita degli Studi di Catania

Citta Universitaria

Viale A. Doria 6

95152 Catania

Italy

**Angelo Gilio**

Dipartimento di Metodi e Modelli Matematici

Via A. Scarpa 16, 00161 Roma (Italy)

**Giuseppe Sanfilippo**

Dipartimento di Scienze Statistiche e Matematiche ''S.Vianelli''

Universita' degli Studi di Palermo

Viale delle Scienze ed.13

90128 Palermo,Italy

** E-mail addresses: **

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