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SSLab 08SEP2020

Statistical Parameters of a Distribution
 The mean of X is a measure of the center of the distribution of X
 The variance of X is the second moment about the mean, and measures the
spread of the distribution of X about the mean
 The third and fourth moments of X about the mean also measure
interesting features of the distribution
 The third moment measures skewness, the lack of symmetry of the
 The fourth moment measures kurtosis, the degree to which the distribution
is peaked
 The actual numerical measures of these characteristics are standardized to
eliminate the physical units, by dividing by an appropriate power of the
standard deviation.
Statistical Parameters
Measure of Skewness
 The distribution X is said to be positively skewed, negatively skewed or
unskewed depending on whether skew(X) is positive, negative, or 0.
 Skewness: indicator used in distribution analysis as a sign of asymmetry and
deviation from a normal distribution.
 In the unimodal case,
 Skewness > 0 - Right skewed distribution - most values are concentrated
on left of the mean, with extreme values to the right.
 Skewness < 0 - Left skewed distribution - most values are concentrated
on the right of the mean, with extreme values to the left.
 Skewness = 0 - mean = median, the distribution is symmetrical around
the mean.
E[( x   x )3 ]
Measure of Skewness
 Relationships of the Mean, Median, and Mode in Symmetrical and Skewed Distributions
Skewness and Kurtosis
Measure of Kurtosis
 Kurtosis is always nonnegative, and in fact is strictly positive unless X is a
 Kurtosis - indicator used in distribution analysis as a sign of flattening or
"peakedness" of a distribution.
E[( x   x ) 4 ]
 In the unimodal case,
Kurtosis > 3 - Leptokurtic distribution, sharper than a normal distribution, with
values concentrated around the mean and thicker tails. This means high probability
for extreme values.
Kurtosis < 3 - Platykurtic distribution, flatter than a normal distribution with a wider
peak. The probability for extreme values is less than for a normal distribution, and
the values are wider spread around the mean.
Kurtosis = 3 - Mesokurtic distribution - normal distribution for example.
 The kurtosis of the standard normal distribution is 3. Using the standard normal
distribution as a benchmark, the excess kurtosis of a random variable X is
defined to be kurt(X)−3. It is defined as excess kurtosis
Rayleigh Distribution
Assuming that each component is uncorrelated, normally distributed with equal
variance, and zero mean, then the overall wind speed (vector magnitude) will be
characterized by a Rayleigh distribution.
Noise Probability Density Functions
PDF’s of noise and signal plus noise
Normal (or Gaussian) distribution