Probability Distribution Functions
The problem introduced in Sect. 1.6.1 with Bertrand’s paradox occurs when we try to decompose the range of possible values of a random variable x into equally probable elementary intervals and this is not always possible without ambiguity because of the continuous nature of the problem. In Sect. 1.6 we considered a continuous random variable x with possible values in an interval [x1, x2], and we saw that if x is uniformly distributed in [x1, x2], a transformed variable y = Y (x) is not in general uniformly distributed in [y1, y2] = [Y (x1), Y (x2)] (Y is taken as a monotonic function of x). This makes the choice of the continuous variable on which equally probable intervals are defined an arbitrary choice.
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