sec-CLT.qmd

## The Central Limit Theorem

The sum of many independent or nearly-independent random variables with
small variances (relative to the number of RVs being summed) 
produces bell-shaped distributions.

For example, consider the sum of five dice (@fig-clt-5d6).

```{r}
#| label: fig-clt-5d6
#| fig-cap: "Distribution of the sum of five dice"
library(dplyr)
dist = 
  expand.grid(1:6, 1:6, 1:6, 1:6, 1:6) |> 
  rowwise() |>
  mutate(total = sum(c_across(everything()))) |> 
  ungroup() |> 
  count(total) |> 
  mutate(`p(X=x)` = n/sum(n))

library(ggplot2)

dist |> 
  ggplot() +
  aes(x = total, y = `p(X=x)`) +
  geom_col() +
  xlab("sum of dice (x)") +
  ylab("Probability of outcome, Pr(X=x)") +
  expand_limits(y = 0)

  
  
```

In comparison, the outcome of just one die is not bell-shaped (@fig-clt-1d6).

```{r}
#| label: fig-clt-1d6
#| fig-cap: "Distribution of the outcome of one die"
library(dplyr)
dist = 
  expand.grid(1:6) |> 
  rowwise() |>
  mutate(total = sum(c_across(everything()))) |> 
  ungroup() |> 
  count(total) |> 
  mutate(`p(X=x)` = n/sum(n))

library(ggplot2)

dist |> 
  ggplot() +
  aes(x = total, y = `p(X=x)`) +
  geom_col() +
  xlab("sum of dice (x)") +
  ylab("Probability of outcome, Pr(X=x)") +
  expand_limits(y = 0)

  
  
```

What distribution does a single die have?

Answer: discrete uniform on 1:6.