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Now let *x*_{t} be a time series made up of identically distributed
random numbers: *m*_{x} and do not depend on time.
Let us also suppose that they are *independently* chosen; this
means in particular that for any different times *t* and *s* ():
| |
(19) |

Suppose we have a sample of *n* points of *x*_{t} and are
trying to determine the value of *m*_{x}.
We could make an estimate of the mean *m*_{x} with the formula
| |
(20) |

A somewhat more elaborate method of estimating the mean
would be to take a weighted average.
Let *w*_{t} define a set of weights normalized so that

| |
(21) |

With these weights, the more elaborate estimate of the mean is
| |
(22) |

Actually (20) is just a special case of (22);
in (20) the
weights are *w*_{t} = 1/*n*.
Further, the weights could be *convolved* on the random time series,
to compute *local* averages of this time series, thus smoothing it.
The weights are simply a filter response where the filter coefficients happen
to be positive and cluster together.
Figure 6 shows an example: a random walk function
with itself smoothed locally.

**walk
**

Figure 6
Random walk and itself smoothed (and shifted downward).

** Next:** Variance of the sample
** Up:** TIME-STATISTICAL RESOLUTION
** Previous:** Probability and independence
Stanford Exploration Project

10/21/1998