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| A stochastic process is a . This means that, if | A stochastic process is a . This means that, if | ||
| f : D -> R | f : D -> R | ||
| is a random function with D and R, the image of each point of D, f(x), is a with values in R. | is a random function with D and R, the image of each point of D, f(x), is a with values in R. | ||
| Of course, the mathematical definition of a includes the case "a function from {1,...,n} to R is a in R^n", so multidimensional random variables are a special case of stochastic processes. | Of course, the mathematical definition of a includes the case "a function from {1,...,n} to R is a in R^n", so multidimensional random variables are a special case of stochastic processes. | ||
| For our first example, take the domain to be N, the , and our range to be R, the . Then, a function f : N -> R is a of real , and the following questions arise: | For our first example, take the domain to be N, the , and our range to be R, the . Then, a function f : N -> R is a of real , and the following questions arise: | ||
| 1) How is a specified? | 1) How is a specified? | ||
| 2) How do we find the answers to typical questions about sequences, such as | 2) How do we find the answers to typical questions about sequences, such as | ||
| a) what is the of the value of f(i)? | a) what is the of the value of f(i)? | ||
| b) what is the that f is ? | b) what is the that f is ? | ||
| c) what is the probability that is f ? | c) what is the probability that is f ? | ||
| d) what is the probability that f(i) has a as i->infty? | d) what is the probability that f(i) has a as i->infty? | ||
| e) if we construct a from f(i), what is the probability that the series ? What is the probability of the sum? | e) if we construct a from f(i), what is the probability that the series ? What is the probability of the sum? | ||
Revision as of 07:36, 6 April 2002
A stochastic process is a . This means that, if
f : D -> R
is a random function with D and R, the image of each point of D, f(x), is a with values in R.
Of course, the mathematical definition of a includes the case "a function from {1,...,n} to R is a in R^n", so multidimensional random variables are a special case of stochastic processes.
For our first example, take the domain to be N, the , and our range to be R, the . Then, a function f : N -> R is a of real , and the following questions arise:
1) How is a specified?
2) How do we find the answers to typical questions about sequences, such as
a) what is the of the value of f(i)?
b) what is the that f is ?
c) what is the probability that is f ?
d) what is the probability that f(i) has a as i->infty?
e) if we construct a from f(i), what is the probability that the series ? What is the probability of the sum?