By Hopf E.

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**Additional resources for On The Time A Verage Theorem In Dynamics**

**Sample text**

2-1). Note that two or more different sample points might give the same value of X(0, but two different numbers in the range cannot be assigned to the same sample point. x (0 Fig. 1 R Random variable X as a function. v. X as (Fig. , say Y or 2, with Y(H)= 0, Y ( T )= 1 or Z ( H ) = 0, Z ( T ) = 0 B. v. and x is a fixed real number, we can define the event (X = x) as (X = x) = {l: X(C) = x) Similarly, for fixed numbers x, x,, and x, , we can define the following events: (X 5 x) = {l: X(l) I x) (X > x) = {C: X([) > x) (xl < X I x2) = {C: XI < X(C) l x2) CHAP.

19). From Eqs. 10),we have for any E 2 0. As F x ( x ) is continuous, the right-hand side of the above expression approaches 0 as Thus, P(X = x ) = 0. 19. v. X is given by 3 O

V. to denote the random variable. Note that the terminology used here is traditional. Clearly a random variable is not a variable at all in the usual sense, and it is a function. v. v. X. Thus the range of X is a certain subset of the set of all real numbers (Fig. 2-1). Note that two or more different sample points might give the same value of X(0, but two different numbers in the range cannot be assigned to the same sample point. x (0 Fig. 1 R Random variable X as a function. v. X as (Fig. , say Y or 2, with Y(H)= 0, Y ( T )= 1 or Z ( H ) = 0, Z ( T ) = 0 B.

### On The Time A Verage Theorem In Dynamics by Hopf E.

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