activity.qmd

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format: pdf
editor: visual
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## Activity {-}

* Think of a random variable constructed by transforming with a certain non-liner function $f$ a categorical, binomial, Poisson or Gaussian random variable.
* Take a sample of size 50, save it into a variable `x`.
* You can calculate the mean of this sample using function `mean()`. Repeat the previous steps different times, for example 1000. Obtaining different means coming from different samples.
* Visualize the means observed in the previous simulation.
* If you only had sample `x` (the initial 50 observations), what would be a reasonable value for $\mathbb{E}[Y]$?
* Using expression
  $$
  \frac{\bar{x} - \mathbb{E}[Y]}{s_Y/\sqrt{n}} \sim t_{n-1}
  $$
  can you think about a region where $\mathbb{E}[Y]$ is likely to be?

For example, the random variable obtained after transforming with the function $sin$ a binomial distribution.

```{r}
set.seed(1)
rv = function(n, size, prob) sin(rbinom(n, size, prob))
x = rv(50, 10, 0.25)
x
```

The mean can be calculated different times for different samples using function `replicate()`.

```{r}
x_means = replicate(1000, mean(rv(50, 10, 0.25)))
```