1.14 Expectation, variance and moments
You can compute and interpret the first two moments of anything you model.
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Expectation and variance are the first two numbers computed about anything random, and linearity of expectation is the most used trick in the subject — it works even when variables are dependent, which surprises everyone once. Covariance and correlation extend the idea to pairs, feeding straight into PCA. It sits here as the measurement layer over distributions. The confusion to retire is forgetting that variance has squared units; standard deviation exists so that the number lives on the same scale as the data.
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Expectation and linearity
Expectation is the long-run average, and its most useful property is that it passes through sums regardless of dependence. That linearity solves problems that look hard.
Variance, covariance, correlation
Variance measures spread, covariance measures how two quantities move together, and correlation is the scale-free version. Nearly every summary statistic in this subject is one of these three.
Skewness and kurtosis in real data
The third and fourth moments describe asymmetry and tail weight, and both matter when data is not well behaved. Ignoring them is how models get surprised by rare events.
Moment generating functions (light touch)
A transform that packages all the moments into one function, touched on lightly here. It is mostly a proof technique, and knowing it exists is enough at this stage.
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