Justify sum steps as distributing the sum over the terms

_subfiles/_def-cond-expectation.qmd

@@ -677,7 +677,7 @@ $$
 \\&= \sum_{y \in \rangef{Y}} \paren{a h_1(x, y) + b h_2(x, y)} \cd \P(Y = y \mid X = x)
 && \text{(definition of conditional expectation)} \\
 &= a \sum_{y \in \rangef{Y}} h_1(x, y) \cd \P(Y = y \mid X = x) + b \sum_{y \in \rangef{Y}} h_2(x, y) \cd \P(Y = y \mid X = x)
-&& \text{(split the sum; factor out the constants)} \\
+&& \text{(distribute the sum over the terms; factor out the constants)} \\
 &= a\E{h_1(X, Y) \mid X = x} + b\E{h_2(X, Y) \mid X = x}
 && \text{(definition of conditional expectation)}
 \ea

_subfiles/_exr-fubini-joint-disc-infinite.qmd

@@ -67,7 +67,7 @@ $$
 \\&= \sum_{x=0}^{\infty} p^x \sb{x(1-p) + p}
 && \text{(multiply } (1-p)^2 \text{ into the parentheses)}
 \\&= (1-p) \sum_{x=0}^{\infty} x\,p^x + p \sum_{x=0}^{\infty} p^x
-&& \text{(split the sum; factor out the constants)}
+&& \text{(distribute the sum over the terms; factor out the constants)}
 \\&= (1-p) \cd \frac{p}{(1-p)^2} + p \cd \frac{1}{1-p}
 && \text{(geometric-series facts)}
 \\&= \frac{p}{1-p} + \frac{p}{1-p}

_subfiles/_exr-fubini-joint-mixed-infinite.qmd

@@ -46,7 +46,7 @@ $$
 \\&= \frac{1-q}{2} \sum_{y=0}^{\infty} (y+1)\,q^y
 && \text{(cancel } y + 1 \text{; factor out } \tfrac{1-q}{2} \text{)}
 \\&= \frac{1-q}{2} \paren{\sum_{y=0}^{\infty} y\,q^y + \sum_{y=0}^{\infty} q^y}
-&& \text{(split the sum)}
+&& \text{(distribute the sum over the terms)}
 \\&= \frac{1-q}{2} \paren{\frac{q}{(1-q)^2} + \frac{1}{1-q}}
 && \text{(geometric-series facts)}
 \\&= \frac{1-q}{2} \cd \frac{q + (1-q)}{(1-q)^2}

_subfiles/_sol-linearity-use.qmd

@@ -1,6 +1,6 @@
 ::: {#sol-linearity-use}
 
-Pull the constant $1/n$ out and split the sum, both by @thm-linearity-expectation:
+Pull the constant $1/n$ out and distribute the sum over the terms, both by @thm-linearity-expectation:
 
 $$
 \begin{aligned}

_subfiles/_thm-lotus.qmd

@@ -130,7 +130,7 @@ $$
 &= a\sum_{x \in \rangef{X}}\sum_{y \in \rangef{Y}} x \cd \P(X=x, Y=y)
  + b\sum_{x \in \rangef{X}}\sum_{y \in \rangef{Y}} y \cd \P(X=x, Y=y)
  + c\sum_{x \in \rangef{X}}\sum_{y \in \rangef{Y}} \P(X=x, Y=y)
-&& \text{(split the sum; constants factor out)}\\
+&& \text{(distribute the sum over the terms; constants factor out)}\\
 &= a\sum_{x \in \rangef{X}}\sum_{y \in \rangef{Y}} x \cd \P(X=x, Y=y)
  + b\sum_{y \in \rangef{Y}}\sum_{x \in \rangef{X}} y \cd \P(X=x, Y=y)
  + c\sum_{x \in \rangef{X}}\sum_{y \in \rangef{Y}} \P(X=x, Y=y)