By Fritschze B.

Sophus Lie (1842-1899) used to be the most very important mathematicians of the 19th century. His paintings on line-sphere transformation and the production of the idea of continuing teams and his software of those to different components of arithmetic was once ground-breaking and has had a long-lasting impression at the additional improvement within the box. certainly, a brand new self-discipline of arithmetic often called Lie idea this present day has resulted. This distinctive mathematician Sophus Lie summarized his lifestyles and paintings within the draft of an highbrow testimonial as follows.

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Un } where ui 2 M for all i. Then M⇤ is countable. But uncountably infinite sets are also relevant, such as with real-valued random variables. A less familiar example is that, if M is a finite set, then the set of all sequences (not just finite sequences) taking values in M is an uncountably infinite set. It turns out that the method adopted thus far to define probabilities over finite sets, namely just to assign nonnegative “weights” to each element in such a way that the weights add up to one, works perfectly well on countable sets.

2 The above observation motivates the notion of the correlation coefficient between two real-valued random variables. 13 Suppose X, Y are real-valued random variables assuming values in finite sets A, B ✓ R respectively. Let denote their joint distribution, and X , Y the two marginal distributions. Let E[XY, ], E[X, X ], E[Y, Y ] denote expectations, and let (X), (Y ) denote the standard deviations of X, Y under their respective marginal distributions. 21) is called the correlation coefficient between X and Y .

Proof. The first part of the lemma says that if the original function h assumes only nonnegative values, then so does its conditional expectation h|X. 33). The second part follows readily upon observing that if h : A ⇥ B ! [↵, ], then both h ↵ and h are nonnegative-valued functions. 2 A very useful property of the conditional expectation is given next. 20 Suppose X, Y are random variables assuming values in finite sets A = {a1 , . . , an }, B = {b1 , . . , bm } respectively. Let 2 Snm denote their joint distribution.

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