Probability and Statistics
- Random phenomena, e.g., Turbulence, Random Acoustic Waves, Random Vibrations, Statistical Thermodynamics and etc.
- Measurement Uncertainty
- Statistical Process Control (SPC) and Design Experiments
andom Phenomena:
- Deterministic Process – Behavior at one or several times determines behavior for all times.
- Example of deterministic process: Suppose we have a wave and further suppose that at t = 0, f (0) = ½, f’(0) > 0
- Random Process – Behavior at any time can be (not necessarily) statistically independent of behavior at any other time. We could say that behavior is only correlated for short times (or spatial distance) and becomes increasingly uncorrelated as the time (or spatial distance) in increased.
- Example of a Random process, Velocity Field in a turbulent jet. Note that the flow is random and these results are not due to random scatter in the data values.
- What Statistical Quantities have physical relevance? Mean Values, Standard Deviations, Probability Density Functions or Histograms (e.g. Gaussian), Spectra and Correlation Functions.
- Mean Values, Time average or Ensemble average
is the estimate of true mean Time Average of Mean Value
Ensemble Average of Mean Value
- Standard Deviations, Time average or Ensemble average
Ensemble Average
Probability Density Functions
- Central Tendency – Tendency towards one central value about which all other values are scattered. I.e., concept of mean value standard deviation and higher moments.
- Probability – Particular interval of values for a random variable are measured at some frequency relative to any other random variable. Leads to the concept of a histogram and frequency distribution. Ultimately leads to probability density function, p(x)
Two examples which demonstrate how to compute a histogram and frequency distribution follow. Example from Text: 4.1 pg. 128 Table from text: 4.1 pg. 127: Sample of variable x Figure from text: 4.2 pg. 128: Histogram and frequency distribution for data in Table 4.1
- Infinite Statistics, assume for now Gaussian or Normal Distribution:
x and s
is the true variance or standard deviation of x. Note that the maximum
will occur at
The probability, We can simplify the integration by transforming to a different set of
variables. If we write ,
as the standardized normal variable for any The value in the square brackets is known as the normal error function and provides ½ the probability over the entire interval. This half value is tabulated in the following table. Table from text: 4.3 pg. 135: Probability Values for Normal Error Example from text: 4.2 pg. 135 Figure from text: 4.3 pg. 134: Integration terminology for the normal error function. Figure from text: 4.4 pg. 136: Relationship
between the probability density function and its statistical parameters
Example from text: 4.3 pg. 137
In real life we deal with finite samples, hence we need to try and quantify
how well we know the mean from the finite sample - We define a sample mean as
and the sample variance as where is called the deviation of - The predictive utility of infinite statistics can be extended to data sets of finite samples size with a few changes. The sample variance can be weighted in such a manner so as to correct for the finite sample of the measured. Lets assume a Normal Distribution and write:
z
variable. The basic problem involves replacing the true standard deviation.
For small N this can be misleading. NOTE:
represents the interval of values in which P% of the measurements
should lie.-
Standard Deviation of the Means – Imagine a though experiment where you
measure a particular variable
*N*times under fixed operating conditions. Now repeat this same experiment*M*times. For each experiment we will obtain value which will in general be different than any other estimate because of the finite sample.
The amount of variation in our individual estimate of the mean should
depend on the sample variance and
the number of samples Figure from text: 4.6 pg. 141: Relationships
between x’.
The standard deviation of the means represents a measure of the precision
in a sample mean. The rang over which the possible values of the true mean
value might lie at some probability level,
so
represents a precision interval at the assigned probability Example from Text: 4.4 pg. 141 Numbers of Measures Required How many measurements, N, are required to give an acceptable precision
in the mean value? The precision interval, CI is estimated by:
The accuracy of this equation depends on how well s. The problem
is that an estimate for the sample variance is needed. A way around this
is to make a few measurements ^{2}N, to obtain an estimate of the sample
variance that we call S. The use _{1}S
to estimate the number of estimates required.
_{1}This establishes that additional measurements will be needed. Example from Text: 4.12 pg. 161 Example from Text: 4.13 pg. 151 |