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I'm struggling to understand some notation in this excerpt from Larsen & Marx. Under "Comment" j is defined as 1, 2,...,r, but the sigma shows the lower bound of j as 0. Is this inconsistency a typo?

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    $\begingroup$ Because $\mu_0=1,$ it never explicitly appears in any formula and does not need to be mentioned. (Nitpick: L&M should have specified that $r$ is non-negative rather than positive. In fact, they didn't need to restrict $r$ at all: the definition makes sense for negative values, real numbers, and even complex numbers, merely by changing $|w|^r$ to $||w|^r|$.) $\endgroup$
    – whuber
    Commented Jan 20, 2023 at 16:03

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No, I think it's correct both times.

That the sum goes from 0 is clear from the binomial expansion. That the comment about what you're writing it in terms of is nevertheless correct can be seen by expanding it out and seeing that the final term involves constants ($\mu_0$ is just $1$) and a power of $\mu_1$; note that the last and second-last terms both only have $\mu_1$.

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Check for yourself:

\begin{align}\mu_r' &:=\mathbb E[(W-\mu)^r]\\ &=\mathbb E\left[W^r-{{r}\choose{ 1}}W^{r-1}\mu^1 + {{r}\choose{ 2}}W^{r-2}\mu^2 - \cdots + (-1)^r\mu^r\right] \\ &= \mu_r -{{r}\choose{ 1}}\mu_{r-1}\mu^1 + {{r}\choose{ 2}}\mu_{r-2}\mu^2 - \cdots + (-1)^r\mu^r.\end{align}

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