# Task 1: Visualize some statistic properties {#1}
## 1. Data
### Generate 500 normally distributed samples
::: callout-tip
You might want to make your results reproducible by fixing some seeds for the random generators. The two most common random generators used in julia are `Random.MersenneTwister` and `StableRNGs.StableRNG` - For this execrise I would recommend the latter (even though MersenneTwister is much more common to be used), thus run:
``` julia
using StableRNGs
randn(StableRNG(1),100)
```
to get 100 random numbers.
:::
Scale the random numbers to fullfill `std(x) ≈ 10`
### functionize it
Next wrap that code in a function `simulate` which takes two arguments, a random seed and the number of samples
## 2. cumulative mean
Calculate the cumulative mean of a single simulation. save it to a variable
Note that there is no `cummean` function, but clever element-wise division in combination with `cumsum` should lead you there - or you just use a loop :shrug:
::: {.callout-tip collapse="true"}
## click to show solution
`cumsum(x) ./ 1:length(x)`
:::
## 3. Plotting!
Now for your first plot. Use a `scatter` plot to visualize the cummulative mean output, if you do not generate a `Figure()` + `ax = f[1,1] = Axis(f)` manually, you can get it back by the scatter call. `f,ax,s = scatter()`. This is helpful as we later want to extend the `Axis` and `Figure` with other plot elements
Use `hlines!` to add a horizontal line at your "true" value
## 4. Subplot
### simulate repeatedly
Let's simulate 1000x datasets, each with a different seed, and take the mean over all simulated values
::: {.callout-tip collapse="true"}
## click to show tip
An easy way to call a function many times is to broadcast it on an array e.g. `1:1000` - you could also use `map` to do it, but I don't think it is as clear :)
:::
::: {.callout-tip collapse="true"}
## click to show solution
`simulate.(1:1000,nmax)`
:::
### Mean it
calculate the mean of each simulation
::: {.callout-tip collapse="true"}
## click to show solution
```julia
using Statistics
mean.(simulate.(1:1000,nmax))
# or
sum.(...) ./ nmax
```
:::
### Add it as a subplot
We want to add a histogram of the 1000 means to the plot.
1. Add a new Axis to `f[1,2]`
2. use it to plot the histogram of the means via `hist!` - don't forget to change the `direction=:x` to flip the histogram
3. link the axes using `linkaxes`
## 5. Prettify it
There are some simple tricks to make a plot look nicer:
- remove the "box" using `hidespines!(ax,:r,:t)
- resize the right sub-plot to be smaller `colsize!` and `Relative(X)`
- hide the x-grid (type `ax.`+ `TAB` to find all possible attributes)
- hide the `xlabels` + `xticks` + `bottomspine` from the right subplot
- add two Labels `(A)` and `(B)` to the plot
- Bonus: use `color` to color the cummulative sum samples according to how many samples went into that sum. `colormap=:Reds` looks good to me!
::: {.callout-tip collapse="true"}
## Bonus: Click for more fancy labels
You can create a slightly fancier label by adding a circle around it :)
Using the `Pluto.jl` reactive backend, changing a value in some cell will automatically update all other cells - including plots.
We can use Sliders instead of fixing the parameters of the simulation
A slider is defined like this:
```julia
@bind yourVarName PlutoUI.Slider(from:to) # from:step:to is optional, step by def 1
```
## Adding interactivity via sliders
1. Define a slider that controls the number of samples from 1:500
2. Define a second slider that adds a constant offset to all values of the simulation simulation
3. make sure to fix the x/y-limits to get a nice looking plot :-)
:::{.callout-tip collapse="true"}
## Bonus: Advanced slider management
After understanding the slightly awkward syntax, the following gives a nice collection of Sliders, Checkboxes, Widgets etc. with at the same time being drag-and-dropable and in a sidebar. Neat!
