Acta Mathematica Academiae Scientiarum Hungaricae 10. (1959)

1959 / 3-4. szám - Rényi A.: On measures of dependence

A. RÉNYI zation of g. If £ is an arbitrary random variable, let cl^ denote the least ст-algebra of subsets of £2 with respect to which £ is measurable. If r\ is another random variable with finite mean value, we denote by M(/;|£) the conditional mean value of ц with respect to a given value of £; M(?;|£) itself is a random variable which is measurable with respect to &£ and is such that for any A £ éli we have (2) |M(#)r/P= |Vp; A A of course, M(ij|£) is unique only if we consider two random variables which are equal with probability 1 to be identical. In what follows we shall always take this for granted. The following two well-known properties (see [3]) of conditional mean values will often be used in the sequel: If M(^) exists, then (3) M(M(7?|g)) = M(7J) and (4) M(^(§)i2||) = g'(g)M(i2||) if g(x) is a Borel-measurable real function of the real variable x. The curve y = M(i?|£ = x) is called the regression curve of t] on £. We shall denote the joint distribution of two random variables £ and г] by Qi,,;, i. e. we put for any Borel subset C of the (x,y)-plane Ое,ч(С) = Р((1,ч)€д where (£, /;) £ C denotes the set of those со £ £2 for which the point with the coordinates £(co), rj(co) belongs to C. We denote, further, by Q;, v the direct product of the distributions of £ and r\, i. e. we put for any two Borel sub­sets A and В of the real line СЬ,(Л*В) Р(^А)Р(/;(В) where A* В denotes the direct product of the sets A and B, i. e. the set of all points (x,y) for which x £ A and у $ В. The definition of is extended to any Borel subset C of the (x, y)-plane in the usual way (see e. g. [4]). § 2. Definitions and fundamental properties of measures of dependence Let £ and ц be random variables on a probability space [£2, ŰL, P], neither of them being constant with probability 1. In almost every field of application of statistics one encounters often the problem that one has to characterize by a numerical value the strength of dependence between £ and ij. Of course, such a value serves only for comparison, and thus its range is

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