chapter3.qmd

{{< include _common.qmd >}}

# Sampling from the normal distribution

## An example

::: {#exm-circuitboard}
Consider the time to failure (in years) for circuit boards modelled by $\Exp(\lambda)$.
An independent random sample $X_1,\dots,X_n$ was collected.
What is the probability that the minimum time lasted is more than 2 years?
\begin{align*}
\Pr(\min\{X_1,\dots,X_n\} > 2)
&=\Pr(X_1 > 2, \dots, X_n > 2) \\
&=\Pr(X_1>2)\cdots\Pr(X_n>2)\\
&= \left[\Pr(X_1 > 2)\right]^n \\
&= [e^{-2/\lambda}]^n \\
&= e^{-2n/\lambda}
\end{align*}
:::


A reasonable estimator for $1/\lambda$ is $T_n=\min\{X_1,\dots,X_n\}$.


## Finite vs inifinite population

[Infinite population]{.ul}

Implicitly iid: "Removing" $X_1=x_1$ from the population does not affect the probability distribution for the subsequent samples. 
Why "infinite"? In scenarios where the exact population size is either unknown, uncountable, or effectively limitless, it is simpler to treat it as infinite.

[Finite population]{.ul}

Not necessarily iid, depending on the sampling method:

- Sampling without replacement
- Cluster sampling
- Stratified sampling
- etc.

Standard errors calculations are affected here.
Check out [Finite Population Correction factors](https://en.wikipedia.org/wiki/Standard_error#Finite_population_correction_(FPC)) if interested.


::: {#exm-infinitesampling}
Estimate the average height of goalkeepers.
Which ones? Presumably all of them--past, present, and future--in all leagues.
For all intents and purposes, this is an infinite population.
:::


## An experiment

Using `R`, we can draw multiple instances of the statistic $T_n$.
Let $n=25$ and $p=0.6$ (true value).

1. Draw $X_1,\dots,X_n\iid \Bern(p)$
2. Compute $T_n = \sum_{i=1}^n X_i$
3. Repeat steps 1--2 a total of $B=10000$ times

```{r}
n <- 25
p <- 0.6
B <- 10000
x <- rbinom(B, n, p)

# First 10 values
head(x, 10)

ggplot() +
  geom_histogram(aes(x), fill = "gray30", col = "white", 
                 breaks = seq(-0.5, n + 0.5, by = 1)) +
  scale_x_continuous(breaks = seq(0, n, by = 5)) +
  labs(x = expression(T[n]), y = "Frequency") 
```