Probability 3
... The expected value or mean of a random variable has special significance. It is the center of mass of the probability measure of the random variable. Definition 1.7 (Mean). The mean or first moment of X is the expected value of X: E (X). If the distribution of a random variable is very heavy tailed, ...
... The expected value or mean of a random variable has special significance. It is the center of mass of the probability measure of the random variable. Definition 1.7 (Mean). The mean or first moment of X is the expected value of X: E (X). If the distribution of a random variable is very heavy tailed, ...
3. CATALAN NUMBERS Corollary 1. cn = 1
... Otherwise, let 2k + 2, 0 ≤ k ≤ n − 1, be the minimum positive integer such that f (2k + 2) = 0 (it is easy to see that such integer must be even). Then g1 (t) = f (t + 1) − 1 for 1 ≤ t ≤ 2k + 1 is a Dyck path of length 2k, and g2 (t) = f (t + 2k + 2) for 0 ≤ t ≤ 2n − 2k − 2 is a Dyck path of length ...
... Otherwise, let 2k + 2, 0 ≤ k ≤ n − 1, be the minimum positive integer such that f (2k + 2) = 0 (it is easy to see that such integer must be even). Then g1 (t) = f (t + 1) − 1 for 1 ≤ t ≤ 2k + 1 is a Dyck path of length 2k, and g2 (t) = f (t + 2k + 2) for 0 ≤ t ≤ 2n − 2k − 2 is a Dyck path of length ...
Constructing Random Times with Given Survival Processes and
... intensity (λu , u ∈ R+ ) of τ under P exists then one can show that for any u ∈ R+ the process (Zt (u)λu Gu , t ∈ [u, ∞)) is a (P, F)-martingale (see El Karoui et al. [3]). This property implies in turn condition (B.2) provided that the intensity process λ does not vanish. ...
... intensity (λu , u ∈ R+ ) of τ under P exists then one can show that for any u ∈ R+ the process (Zt (u)λu Gu , t ∈ [u, ∞)) is a (P, F)-martingale (see El Karoui et al. [3]). This property implies in turn condition (B.2) provided that the intensity process λ does not vanish. ...
Patterns and Sequences
... The patterns is to subtract 11 to the previous term. The next two terms are 6-11= -5 and -5-11 = -16 ...
... The patterns is to subtract 11 to the previous term. The next two terms are 6-11= -5 and -5-11 = -16 ...
Continued fraction expansion of the square-root operator
... Continued fractions first appeared in the works of the Indian mathematician Aryabhata in the 6th century. He used them to solve linear equations. They re-emerged in Europe in the 15th and 16th centuries and Fibonacci attempted to define them in a general way. The term "continued fraction" first appe ...
... Continued fractions first appeared in the works of the Indian mathematician Aryabhata in the 6th century. He used them to solve linear equations. They re-emerged in Europe in the 15th and 16th centuries and Fibonacci attempted to define them in a general way. The term "continued fraction" first appe ...
Lecture_Notes (reformatted)
... problems to and from which we are reducing. To show a problem is hard, reduce a hard problem to it. To show a problem is easy, reduce it to an easy problem. This is why we choose the ≤ symbol to indicate A ≤ B when a problem A reduces to a problem B. Example: Theorem Proving by Resolution: Mechanica ...
... problems to and from which we are reducing. To show a problem is hard, reduce a hard problem to it. To show a problem is easy, reduce it to an easy problem. This is why we choose the ≤ symbol to indicate A ≤ B when a problem A reduces to a problem B. Example: Theorem Proving by Resolution: Mechanica ...
Lecture 22 - Modeling Discrete Variables
... Random variable – A variable whose value is determined by the outcome of a procedure. The procedure includes at least one step whose outcome is left to chance. Therefore, the random variable takes on a new value each time the procedure is performed, even though the procedure is exactly the same. ...
... Random variable – A variable whose value is determined by the outcome of a procedure. The procedure includes at least one step whose outcome is left to chance. Therefore, the random variable takes on a new value each time the procedure is performed, even though the procedure is exactly the same. ...
An extension of the square root law of TCP
... with probability 1 − e−αWn . This paper consists of three main parts. The first part (see Section 2) is devoted to analyzing the macroscopic behavior of the Markov chain {Wn } as α ↓ 0. We identify the appropriate space scale and time scale that prevail when the loss rate α ↓ 0. This leads to a limi ...
... with probability 1 − e−αWn . This paper consists of three main parts. The first part (see Section 2) is devoted to analyzing the macroscopic behavior of the Markov chain {Wn } as α ↓ 0. We identify the appropriate space scale and time scale that prevail when the loss rate α ↓ 0. This leads to a limi ...
