Conditional expectation
... Let pΩ, F, P q be a probability space and X an integrable random variable. Let Y be a discrete random variable taking values in a finite set S “ tk1 , . . . , kn u, and assume that P pY “ kj q ą 0 for all j. Define f pkq “ ErX | Y “ ks for all k P S. Since tY “ ku is an event with positive probabili ...
... Let pΩ, F, P q be a probability space and X an integrable random variable. Let Y be a discrete random variable taking values in a finite set S “ tk1 , . . . , kn u, and assume that P pY “ kj q ą 0 for all j. Define f pkq “ ErX | Y “ ks for all k P S. Since tY “ ku is an event with positive probabili ...
3 Lipschitz condition and Lipschitz continuity
... (c) If f is Lipschitz continuous on all of the space Rm (i.e. B = Rm in (15)), then f is called globally Lipschitz-continuous. 3.2 Remark (local vs. global) Notice the fundamental difference between the local and global versions of the Lipschitz-continuity. Whereas in the local version the Lipschitz ...
... (c) If f is Lipschitz continuous on all of the space Rm (i.e. B = Rm in (15)), then f is called globally Lipschitz-continuous. 3.2 Remark (local vs. global) Notice the fundamental difference between the local and global versions of the Lipschitz-continuity. Whereas in the local version the Lipschitz ...
On Exhaustion of Domains - Department of Mathematics, Statistics
... therefore for the question of exhaustion one can consider the whole class {G1 } of domains biholomorphically equivalent to G1 . We now introduce the set E(D) of all classes {G}, where G is a bounded domain in Cn that can be exhausted by D. Obviously E(D) is not empty: it always contains {D}. By #E(D ...
... therefore for the question of exhaustion one can consider the whole class {G1 } of domains biholomorphically equivalent to G1 . We now introduce the set E(D) of all classes {G}, where G is a bounded domain in Cn that can be exhausted by D. Obviously E(D) is not empty: it always contains {D}. By #E(D ...
Full text
... irrationality of e. However, as a consequence of the approach taken, one needed to restrict attention to those sequences {£/„}, generated with respect to the relatively prime pair (P,Q) with 121 = 1 and | P | > 1. In view of these results it was later conjectured in [7] whether other irrational valu ...
... irrationality of e. However, as a consequence of the approach taken, one needed to restrict attention to those sequences {£/„}, generated with respect to the relatively prime pair (P,Q) with 121 = 1 and | P | > 1. In view of these results it was later conjectured in [7] whether other irrational valu ...
Chapter 7- counting techniques
... by using pigeonhole principle. Sol: Because there are 8 people and only 7 days per week, so Pigeonhole Principle says that, at least two or more people were being born in the same day. Note that Pigeonhole Principle provides an existence proof. There must be an object or objects with certain cha ...
... by using pigeonhole principle. Sol: Because there are 8 people and only 7 days per week, so Pigeonhole Principle says that, at least two or more people were being born in the same day. Note that Pigeonhole Principle provides an existence proof. There must be an object or objects with certain cha ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.