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concrete mathematics a foundation for computer science phần 7

thử nghiệm độc lập trong một cách mà mỗi giá trị của X xảy ra với một tần số xấp xỉ tỷ lệ thuận với xác suất của nó. (Ví dụ, chúng ta có thể cuộn một đôi lần con xúc xắc nhiều, quan sát các giá trị của S và / hoặc P.) | 8.1 DEFINITIONS 371 making independent trials in such a way that each value of X occurs with a frequency approximately proportional to its probability. For example we might roll a pair of dice many times observing the values of s and or p. We d like to define the average value of a random variable so that such experiments will usually produce a sequence of numbers whose mean median or mode is approximately the same as the mean median or mode of X according to our definitions. Here s how it can be done The mean of a random real-valued variable X on a probability space fi is defined to be Ỵ x-Pr X x 8.6 xex O if this potentially infinite sum exists. Here X n stands for the set of all values that X can assume. The median of X is defined to be the set of all X such that Pr X x 3 ị and Pr X x ỷ 1 2 2 And the mode of X is defined to be the set of all X such that Pr X x Pr X x for all x e X n . 8.7 8.8 In our dice-throwing example the mean of s turns out to be 2 Jg 3 12- JỊ 7 in distribution Pr00 and it also turns out to be 7 in distribution Pn 1. The median and mode both turn out to be 7 as well in both distributions. So s has the same average under all three definitions. On the other hand the p in distribution Proo turns out to have a mean value of Y 12.25 its median is 10 and its mode is 6 12 . The mean of p is unchanged if we load the dice with distribution Pr 11 but the median drops to 8 and the mode becomes 6 alone. Probability theorists have a special name and notation for the mean of a random variable They call it the expected value and write EX 22 X w PrM 8-9 In our dice-throwing example this sum has 36 terms one for each element of 0 while 8.6 is a sum of only eleven terms. But both sums have the same value because they re both equal to 2 xPr a x X cu even XẼX Õ 372 DISCRETE PROBABILITY The mean of a random variable turns out to be more meaningful in applications than the other kinds of averages so we shall largely forget about medians and modes from now on. We