By Edwin P. Alexander

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**Extra resources for American Locomotives, 1900-1950: A Pictorial Record of Steam Power**

**Sample text**

12) and a''= Note. 13) denotes a spat;e variable rather than time the process is called homogeneous. Note. The process is called stationary ~n the strict sense if its entire statistics remains unchanged under a time-shift Chap. I - Elements of Probability Theory - 30 for each n . (3 .. 14) holds for n. ~ k only rather than for all n. the process is called stationary of order k • A pro- cess of order 2 is also weakly stationary; the con- verse is not true! 4. Transformation of a Random Process .

If m is independent of depends on the difference is called (weakly) t-5 t , and if R. (t,5) only the random process stationary. ~'is then a constant. 12) and a''= Note. 13) denotes a spat;e variable rather than time the process is called homogeneous. Note. The process is called stationary ~n the strict sense if its entire statistics remains unchanged under a time-shift Chap. I - Elements of Probability Theory - 30 for each n . (3 .. 14) holds for n. ~ k only rather than for all n. the process is called stationary of order k • A pro- cess of order 2 is also weakly stationary; the con- verse is not true!

Comes ~ 1 and 'I (t ,~i;) hence, one time-function If, however, two out- t;2. exist such that btrl: we speak of a random operator. If, for instance, T is a differential operator then it is deterministic or random depending on whether its coefficients are deterministic or random, respectively. For the time be- ing we restrict ourselves to deterministic operators*) and consider two special cases. (a) Time- Invariant Systems. Let the operation be a function of X but not of X or Jx (r) t d:t etc. 0 Then the output y (t1)at time t1 T(x) depends on :X.