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| A ] is a ]. This means that, if | A ] 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: | ||
| # How is a ] specified? | # How is a ] specified? | ||
| # How do we find the answers to typical questions about sequences, such as | # How do we find the answers to typical questions about sequences, such as | ||
| ## what is the ] of the value of f(i)? | ## what is the ] of the value of f(i)? | ||
| ## what is the ] that f is ]? | ## what is the ] that f is ]? | ||
| ## what is the probability that is f ]? | ## what is the probability that is f ]? | ||
| ## what is the probability that f(i) has a ] as i->infty? | ## what is the probability that f(i) has a ] as i->infty? | ||
| ## if we construct a ] from f(i), what is the probability that the series ]? What is the probability ] of the sum? | ## if we construct a ] from f(i), what is the probability that the series ]? What is the probability ] of the sum? | ||
Revision as of 07:46, 6 April 2002
A stochastic process is a random function. This means that, if
f : D -> R
is a random function with domain D and range R, the image of each point of D, f(x), is a random variable with values in R.
Of course, the mathematical definition of a function includes the case "a function from {1,...,n} to R is a vector in R^n", so multidimensional random variables are a special case of stochastic processes.
For our first infinite example, take the domain to be N, the natural numbers, and our range to be R, the real numbers. Then, a function f : N -> R is a sequence of real numbers, and the following questions arise:
- How is a random sequence specified?
- How do we find the answers to typical questions about sequences, such as
- what is the probability distribution of the value of f(i)?
- what is the probability that f is bounded?
- what is the probability that is f monotonic?
- what is the probability that f(i) has a limit as i->infty?
- if we construct a series from f(i), what is the probability that the series converges? What is the probability distribution of the sum?