Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics. Patrick Muldowney
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A sample of 100 individuals is selected, their individual weights are measured, and the results are summarized in Table 2.2.Estimate the mean weight and standard deviation of the weights in the sample.
Table 2.2 Relative frequency table of distribution of weights.
Weights (kg) | Proportion of sample |
---|---|
0 – 20 | 0.2 |
20 – 40 | 0.3 |
40 – 60 | 0.2 |
60 – 80 | 0.2 |
80 – 100 | 0.1 |
Table 2.3 Calculation of mean and standard deviation.
|
|
x |
|
|
|
---|---|---|---|---|---|
0 – 20 | 0.2 | 10 | 100 | 2 | 20 |
20 – 40 | 0.3 | 30 | 900 | 9 | 270 |
40 – 60 | 0.2 | 50 | 2500 | 10 | 500 |
60 – 80 | 0.2 | 70 | 4900 | 14 | 980 |
80 – 100 | 0.1 | 90 | 8100 | 9 | 810 |
Sometimes calculation of the mean and standard deviation is done by setting out the workings as in Table 2.3. The observed weights of the sample members are grouped or classified in intervals
while the variance of the weights is approximately
The latter calculation, involving
are approximations to the Stieltjes (or Riemann–Stieltjes) integrals
the domain of integration [0,100] being denoted by
In Section 2.1 the variables are discrete. But the outcomes there can be expressed as discrete elements of a continuous domain provided the probabilities are formulated as atomic functions on the domain.
In contrast, the variables