Moment Generating Function (MGF) & Probability Generating Function (PGF)Ĭontents (Click to skip to that section): We can write this as μ 2‘= EX 2, but usually we just write it as σ 2. This happens to be the variance of our distribution. This is just the mean of the distribution. We’d write this simply as μ, and we can write μ = E(X). When r = 1, we are looking at the first moment of a distribution X.Sometimes they are calculated from the definition other times they are calculated using an MGF. In physics, all moments are used, including higher-order n. The mean tells us what the average values look like, and the variance tells us about the spread. In calculus based statistics, the first two moments of distribution are most important. Low 5th = more change in the shoulders.A high 5th = heavy tail, little mode movement.Higher-order terms (above the 4th) are difficult to estimate and equally difficult to describe in layman’s terms.įor example, the 5th r measures the relative importance of tails versus center (mode, shoulders) in the cause of a distribution’s skew. Negative kurtosis = not much data in your tails.
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