Split four multi-div subfiles into one div per file (wave 1)

_subfiles/_cor-best-predictor-argmin.qmd

@@ -1,30 +1,5 @@
-:::{#cor-best-predictor-argmin}
-#### Conditional mean as the minimizer of squared-error risk
-
-Let $\Predvar$ and $\Outvar$ be [jointly distributed](random-variables.qmd#def-jointly-distributed) random variables with $\E{\Outvar^2} < \infty$.
-Among [prediction functions](#def-prediction) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$,
-the [conditional expectation function](#def-cond-expectation-function) is a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer) of the [squared-error risk](#def-squared-error-risk):
-
-$$\E{\Outvar \mid \Predvar} \in \argmin_{\predfn} \E{\sqf{\Outvar - \predfnf{\Predvar}}},$$
-
-where $\argmin$ is the [set of minimizers](https://morrison-lab.github.io/mds/algebra-basics.html#def-argmin).
-
-:::
-
-::: proof
-
-By @thm-best-predictor, $\bestpredfnf{\Predvar} = \E{\Outvar \mid \Predvar}$ has [risk](#def-risk) no larger than that of any [prediction function](#def-prediction) $\predfn$,
-so it is a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer).
-
-:::
+{{< include pds/_subfiles/cor-best-predictor-argmin/_cor-best-predictor-argmin.qmd >}}
 
 {{< slidebreak >}}
 
-:::{#exm-best-predictor-argmin}
-#### The minimizer in the best-predictor exercise
-
-In @exr-best-predictor, the [conditional expectation function](#def-cond-expectation-function) $\bestpredfnf{\Predvar}$ has [risk](#def-risk) $1$.
-The constant prediction $\predfn_1\paren{\Predvar} = 3$ has risk $5$ and $\predfn_2\paren{\Predvar} = 4\Predvar$ has risk $2$, both larger than $1$.
-So $\bestpredfn$ is the [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer) among the three.
-
-:::
+{{< include pds/_subfiles/cor-best-predictor-argmin/_exm-best-predictor-argmin.qmd >}}

_subfiles/_cor-best-predictor-unique.qmd

@@ -1,29 +1,5 @@
-:::{#cor-best-predictor-unique}
-#### The minimizer of squared-error risk is unique up to probability-1 equality
-
-Let $\Predvar$ and $\Outvar$ be [jointly distributed](random-variables.qmd#def-jointly-distributed) random variables with $\E{\Outvar^2} < \infty$.
-Every [prediction function](#def-prediction) $\predfnf{\Predvar}$ with $\E{\predfnf{\Predvar}^2} < \infty$
-that is a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer) of the [squared-error risk](#def-squared-error-risk)
-(see @cor-best-predictor-argmin)
-satisfies:
-
-$$\P(\predfnf{\Predvar} = \E{\Outvar \mid \Predvar}) = 1$$
-
-:::
-
-::: proof
-
-If $\predfn$ is a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer), its [risk](#def-risk) equals that of $\bestpredfn$,
-so the equality case of @thm-best-predictor gives $\P(\predfnf{\Predvar} = \bestpredfnf{\Predvar}) = 1$.
-
-:::
+{{< include pds/_subfiles/cor-best-predictor-unique/_cor-best-predictor-unique.qmd >}}
 
 {{< slidebreak >}}
 
-:::{#exm-best-predictor-unique}
-#### Uniqueness in the best-predictor exercise
-
-In @exr-best-predictor, $\predfn_2\paren{\Predvar} = 4\Predvar$ differs from $\bestpredfnf{\Predvar}$ when $\Predvar = 0$, which has probability $0.5$.
-So $\predfn_2$ is not a [minimizer](https://morrison-lab.github.io/mds/algebra-basics.html#def-global-minimizer), which matches its [risk](#def-risk) $2$ being larger than $1$.
-
-:::
+{{< include pds/_subfiles/cor-best-predictor-unique/_exm-best-predictor-unique.qmd >}}

_subfiles/_def-binomial.qmd

@@ -1,29 +1,3 @@
-:::{#def-binomial}
-#### Binomial distribution
+{{< include pds/_subfiles/def-binomial/_def-binomial.qmd >}}
 
-A [discrete](random-variables.qmd#def-discrete-rv) random variable $X$
-has the **binomial distribution**
-with number of trials $n \in \set{1, 2, \dots}$
-and event probability $\prob \in [0, 1]$,
-written $X \sim \dBinf{n, \prob}$
-[@wikipedia_binomial_distribution],
-if:
-
-$$
-\P(X = x) \eqdef \binom{n}{x} \prob^x (1 - \prob)^{n - x},
-\quad x \in \set{0, 1, \dots, n}
-$$ {#eq-binom-pmf}
-
-where $\binom{n}{x} = \frac{n!}{x!\,(n - x)!}$
-is the number of ways to choose $x$ of the $n$ trials.
-
-:::
-
-::: {.remark .notes}
-
-@CaseBerg01 [Section 3.2] defines the binomial distribution the same way.
-With $n = 1$, @eq-binom-pmf is the
-[Bernoulli PMF](random-variables.qmd#def-bernoulli):
-$\dBinf{1, \prob}$ and $\Ber(\prob)$ are the same distribution.
-
-:::
+{{< include pds/_subfiles/def-binomial/_def-binomial-remark-1.qmd >}}