SC0x_03_Statistics.jl

### A Pluto.jl notebook ###
# v0.19.5

using Markdown
using InteractiveUtils

# ╔═╡ bb8410d0-b941-11ec-3e9e-ff5dc042c03e
begin
	using HypertextLiteral,PlutoUI,LaTeXStrings, Formatting
	
	using Logging
	Logging.disable_logging(Logging.Info) # Disables printing on the (pluto) terminal
	
	using Plots,Plots.PlotMeasures # To be able to use cm, mm, etc.
	default(legend=false,
			size=(250,225),
			tickfontsize=8,
			line=(:2, :blue),
			framestyle=:origin
	)
	
	html"""
	Notebook Packages & Settings
	<style> main {max-width: 925px; } </style>
	<style> .rich_output {color: black;}</style>
	"""	
end

# ╔═╡ 56955eab-e091-4aa8-a42b-7ee5ebb86a25
function center_div(something)
	@htl "<div align='center'>$something</div>"
end;

# ╔═╡ 74b3646b-b73f-4b3c-9006-2b7fd4a01ab1
md"""$\require{physics}$"""

# ╔═╡ 9250300b-7154-40b8-8625-f8c7d0252b2a
TableOfContents(title="Statistics")

# ╔═╡ a9d74a5b-c228-4389-a55e-1bbbb8ae988a
md"""
# Properties of Random Variables
## Basic Definitions
!!! warning ""
	| | |
	|:--|:--|
	|$X$|Random variable that follows a probability distribution|
	|$iid$|"Independent and Identically Distributed" (random variables)|
	|$x_1, x_2,...$|Realizations of the random variable|

"""

# ╔═╡ a122646f-4f99-4f2d-bd79-e4f5fc998010
md"""
## Linearity of the Expectation
Let $X$ be a random variable and $a,b \in \mathbb{R}$, then

$\begin{align}
E[aX + b] &= \sum_{i=1}^{n}p_x(ax_i + b) \\
	&= a\sum_{i=1}^{n}p_ix_i + \sum_{i=1}^{n}p_ib
\end{align}$

!!! warning ""
	$E[aX + b] = aE[X] + b$

and

$\begin{align}
Var[aX + b] &= E\left[(aX + b - E[aX + b])^2\right] \\
	&= E\left[(aX + b- aE[X] - b)^2\right] \\
	&= E[a^2(X-E[X])^2]
\end{align}$

!!! warning ""
	$Var[aX + b] = a^2Var[X]$

"""

# ╔═╡ 624cbd17-40a7-4f37-bae8-558e2bb0b6dc
md"""
## Expectation of a Sum
- Let $X_1, X_2$ be i.i.d. random variables that can take values $x_i$ for $i=1...n$ and $x_j$ for $j=1...n$, respectively
- Let $P(X_1=x_i \cap X_2 = x_j)$ the probability that both realizations $x_i$ and $x_j$ occur simultaneously

$\begin{align}
E[X_1 + X_2] &= \sum_{i=1}^{n}\sum_{j=1}^{n}P(X_1=x_i \cap X_2=x_j)(x_i+x_j) \\
	&= \sum_{i=1}^{n}\sum_{j=1}^{n}P(X_1=x_i \cap X_2=x_j)x_i + 
		\sum_{i=1}^{n}\sum_{j=1}^{n}P(X_1=x_i \cap X_2=x_j)x_j
\end{align}$

Note that
-  $\displaystyle\sum_{\alpha}\sum_{\beta}\alpha\beta = \alpha_1(\beta_1+...)+\alpha_2(\beta_1+...)+...=\sum_{\alpha}\alpha\sum_{\beta}\beta$
- Since $\quad\displaystyle\sum_{\alpha}\sum_{\beta}\alpha\beta\quad$ is a combinatorial sum, we can also write it as $\quad\displaystyle\sum_{\beta}\sum_{\alpha}\beta\alpha$
- For independent variables $P(\alpha \cap \beta) = P(\alpha)P(\beta)$
-  $\displaystyle\sum_{i=1}^{n}P(\alpha_i) = 1$
(estas aclaraciones con respecto a sumatorias deberían ir en una notebook aparte)
> Combinatorics/summation_rules.jl
, then

$\begin{align}
E[X_1 + X_2] &= \sum_{i=1}^{n}x_i\sum_{j=1}^{n}P(X_1=x_i) P(X_2=x_j) + 
		\sum_{j=1}^{n}x_j\sum_{i=1}^{n}P(X_1=x_i) P(X_2=x_j) \\
	&= \sum_{i=1}^{n}x_i P(X_1=x_i) + \sum_{j=1}^{n}x_j P(X_2=x_j) \\
	&= E[X_1] + E[X_2]
\end{align}$

This can be generalized to

!!! warning ""
	$E\left[\sum_{i=1}^{n}X_i\right]=\sum_{i=1}^{n}E[X_i]$

"""

# ╔═╡ d07d83de-0473-4f55-93cb-a5d976882418
md"""
## Expectation of a Product
Let $X_1, X_2$ be $iid$ random variables

$\begin{align}
E[X_1X_2] &= \sum_{i=1}^{n}\sum_{j=1}^{n}P(X_1=x_i \cap X_2=x_j)(x_ix_j) \\
	&= \sum_{i=1}^{n}\sum_{j=1}^{n}P(X_1=x_i) P(X_2=x_j)(x_i)(x_j) \\
	&= \sum_{i=1}^{n}P(X_1=x_i)x_i\sum_{j=1}^{n}P(X_2=x_j)x_j
\end{align}$

!!! warning ""
	$E[X_1X_2] = E[X_1]E[X_2]$
"""

# ╔═╡ 9de3cfdf-c499-4d64-bc4b-6efa98537fb3
md"""
## Expectation of the Expectation
$\begin{align}
E\big[E[X]\big] &= \sum_{i=1}^{n}p_iE[X]
\end{align}$

!!! warning ""
	$E\big[E[X]\big] = E[X]$
"""

# ╔═╡ e7f4bb7f-1f56-4b1a-9609-2f2e60c49ce8
md"""
## Covariance
Defined as


$\begin{align}
Cov(X,Y) &= E\Big[(X-E[X])(Y-E[Y])\Big] \\
	&= E\Big[XY - XE[Y] - E[X]Y + E[X]E[Y]\Big] \\
	&= E[XY] - E[X]E[Y] - E[X]E[Y] + E[X]E[Y]
\end{align}$

!!! warning ""
	$Cov(X,Y) = E[XY] - E[X]E[Y]$

If $X,Y$ are $iid$ random variables, then

$E[XY] = E[X]E[Y] \implies Cov(X,Y)=0$

"""

# ╔═╡ 027fc87f-652c-41e8-ad77-84d8a223fedd
md"""
## Variance of a Sum
Let $X,Y$ be random variables, then

$\begin{align}
Var[X + Y] &= E\big[(X + Y)^2\big] - \big(E[X + Y]\big)^2 \\
	&= E\big[X^2 + 2XY + Y^2\big] - \big(E[X] + E[Y]\big)^2 \\
	&= E[X^2] + 2E[XY] + E[Y^2] - E^2[X] - 2E[X]E[Y] - E^2[Y] \\
	&= Var[X] + Var[Y] + 2\big(E[XY] - E[X]E[Y]\big)
\end{align}$

!!! warning ""
	$Var[X+Y] = Var[X] + Var[Y] + 2Cov(X,Y)$
	
For $iid$ random variables we can extend the property to

!!! warning ""
	$Var\left[\sum_{i=1}^{n}X_i\right] = \sum_{i=1}^{n}Var[X_i]$

"""

# ╔═╡ 3eec3ecc-b526-4010-a4f8-f6ee22a6c2c2
md"""
# Central Limit Theorem
Let $X_1,X_2,...,X_n$ be $iid$ random variables that follow a normal distribution $N(\mu, \sigma)$, then

| | |
|:--|:--|
|Sum of $n$ random variables|$S_n = \displaystyle\sum_{i=1}^{n}X_i$|
|Mean of $n$ random variables|$\bar{X}_n=\displaystyle\frac{S_n}{n}$|

## Sum distribution

Now, if $n$ is large enough, e.g. $>30$, (sometimes > $40$ is needed), then the sum and the mean follow a normal distribution, with mean

$\begin{align}
E[S_n] &= E\left[\sum_{i=1}^{n}X_i\right] \\
	&= \sum_{i=1}^{n}E[X_i]
\end{align}$

!!! warning ""
	$E[S_n] = n\mu$

and variance/standard deviation

$\begin{align}
Var[S_n] &= Var\left[\sum_{i=1}^{n}X_i\right] \\
	&= \sum_{i=1}^{n}Var[X_i] \\

\end{align}$

!!! warning ""
	$Var[S_n] = n\sigma^2$

!!! warning ""
	$\sigma_{S_n} = \sqrt{n}\sigma$
"""

# ╔═╡ 7314ed82-0871-493b-8af2-11be4e0b8077
md"""
## Mean distribution

$\begin{align}
E[\bar{X}_n] &= E\left[\frac{S_n}{n}\right] \\
	&= \frac{1}{n}E\left[\sum_{i=1}^{n}X_i\right]
\end{align}$

!!! warning ""
	$E[\bar{X}_n] = \mu$

$\begin{align}
Var[\bar{X}_n] &= E\left[\left(\frac{S_n}{n}\right)^2\right] - \left(E\left[\frac{S_n}{n}\right]\right)^2 \\
	&= \frac{1}{n^2}\left(E[(S_n)^2] - (E[S_n])^2\right) \\
	&= \frac{1}{n^2}Var[S_n] \\
	&= \frac{1}{n^2}n\sigma^2 
\end{align}$

!!! warning ""
	$Var[\bar{X}_n] = \frac{\sigma^2}{n}$

!!! warning ""
	$\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$ 
"""

# ╔═╡ 4898cf5b-50d2-4dc6-9c3d-6195c5ba21a3
md"""
# Sampling & Confidence Intervals

!!! warning ""
	| | | |
	|:--|:--|:--|
	|$\bar{X}$|Random variable that follows a probability distribution|e.g Theoretical sample mean |
	|$\bar{x}, \bar{x_1}...$|Realizations of the previous random variable| e.g. Observed sample mean|

## Known variance
- Let $X_1,...,X_n$ be a random sample from a normal distribution $\sim N(\mu, \sigma)$
- Then $\bar{X}$ is also a random variable and (if $n$ is large enough, see Central Limit Theorem) $\bar{X} \sim N\left(\mu, \sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\right)$

- Note that if we standardize $\bar{X}$, i.e. $\displaystyle Z= \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$ we get $Z \sim N(0,1)$.
- Choose $-z$ and $z$ such that $P(-z<Z<z)=p$ for a $Z \sim N(0,1)$, e.g. 
  +  $p=0.90 \implies -z \approx 1.64, z \approx 1.64$
  +  $p=0.95 \implies -z \approx 1.96, z \approx 1.96$

Now, note that

$\begin{align}
p &= P\left( -z < \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} < z\right) \\
	&= P\left(-z\frac{\sigma}{\sqrt{n}} < \bar{X} - \mu < z\frac{\sigma}{\sqrt{n}} \right) \\
	&= P\left(- \bar{X} -z\frac{\sigma}{\sqrt{n}}  < - \mu < -\bar{X} + z\frac{\sigma}{\sqrt{n}} \right) \\
	&= P\left(\bar{X} - z\frac{\sigma}{\sqrt{n}} < \mu < \bar{X} + z\frac{\sigma}{\sqrt{n}} \right)
\end{align}$

In some cases we will have $n<30$ sample sizes, so we won't be calculating intervals using a normal distribution, but a t-distribution. So let's change the exclusive notation $z$ (or $t$) to $c$, so both cases are implied. Then, we've found that 

!!! warning ""
	The interval
	
	$\left[\bar{X} - c\frac{\sigma}{\sqrt{n}},  \bar{X} + c\frac{\sigma}{\sqrt{n}} \right]$
	
	covers $\mu$ with a probability $p$.

**Example:**

In the quality control study of a particular product, a sample of $n=23$ items is collected. Previous studies show that a correct manufacturing process have $\sigma=0.1$. Compute a $95\%$ confidence interval for the true mean ($\mu$) knowing that the sample mean is $\bar{x}=72.886$

- Interval: $\left[72.886 - 1.96\frac{0.1}{\sqrt{23}}, 72.886 + 1.96\frac{0.1}{\sqrt{23}}\right] = [72.845, 72.927]$

"""

# ╔═╡ a72ad4c1-1c89-475e-9638-0188bd19a006
md"""
## Unknown variance
If the population variance (e.g. distribution variance) is unknown, we have to estimate the bounds of the intervald. Let $s$ be the sample standard deviation, then

!!! warning ""
	The interval
	
	$\left[\bar{X} - c\frac{s}{\sqrt{n}},  \bar{X} + c\frac{s}{\sqrt{n}} \right]$
	
	estimates $\mu$ with a probability $p$.

"""

# ╔═╡ 1cf38e77-ce5e-457e-a8e9-cd44a544011b
md"""
## Interpretation
- Typically, sampling is executed just once and a confidence level $p=1-\alpha$ is reported
- If we repeat the sampling procedure $100$ times, these random intervals will capture the true population mean, on average, $p(100)$ times out of $100$.
"""

# ╔═╡ 134ea9f0-bd47-4d23-8dac-f5ac4c00c02c
md"""
# 🔨 Hypothesis Testing
## Intro
Hypothesis testing is used when we want to test if a measurement is significantly different, smaller or greater than a hypothesized value.
- Let $X_T$ be a random variable.
- Let $x_T$ be a realization/observation of that random variable.
- We want to test whether $x_T$ differs from a hypothetical value $h$
- Let $H_0$ be the null hypothesis. Let $H_1$ be the alternative hypothesis.

Then we have the following cases

| | |
|:--|:--|
|Shorter|$\begin{align}H_0: x_T \geq h \\ H_1: x_T < h \end{align}$|
| | |
|Different|$\begin{align}H_0: x_T = h \\ H_1: x_T \neq h \end{align}$|
| | |
|Greater|$\begin{align}H_0: x_T \leq h \\ H_1: x_T > h \end{align}$|

**Common Procedure**

| | | |
|:--|:--|:--|
|1|Test statistic selection|Sum, average, error, etc.|
|2|Determine whether is a 1 or 2 tailed test|? $\begin{cases}= & 2 \text{ tail} \\ > or < & 1 \text{ tail} \end{cases}$|
|3|Pick a significance level & critical value|c = $\begin{cases}CDF^{-1}(1-\alpha) & 2 \text{ tail} \\ CDF^{-1}(1-\frac{\alpha}{2}) & 1 \text{ tail} \end{cases}$|
|4| Formulate Null & Alternative hypotheses|
|5| Calculate the test statistic| $\begin{cases}\text{An interval} & 2 \text{ tail} \\ \text{A point} & 1 \text{ tail} \end{cases}$
|6| Comparison
"""

# ╔═╡ afc8d58c-65c8-4945-b52d-b99c6e91c719
md"""
# 🔨 Multiple Random Variables
"""

# ╔═╡ 988ecb55-6d61-4502-960a-afec6fdf1b36
md"""
# 🔨 Regression
"""

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[[deps.Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[deps.Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[deps.Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[deps.SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "4f6ec5d99a28e1a749559ef7dd518663c5eca3d5"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.4.3"

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "c3d8ba7f3fa0625b062b82853a7d5229cb728b6b"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.2.1"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "8977b17906b0a1cc74ab2e3a05faa16cf08a8291"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.16"

[[deps.StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "57617b34fa34f91d536eb265df67c2d4519b8b98"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.5"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[deps.TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits", "Test"]
git-tree-sha1 = "5ce79ce186cc678bbb5c5681ca3379d1ddae11a1"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.7.0"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─bb8410d0-b941-11ec-3e9e-ff5dc042c03e
# ╟─56955eab-e091-4aa8-a42b-7ee5ebb86a25
# ╟─74b3646b-b73f-4b3c-9006-2b7fd4a01ab1
# ╠═9250300b-7154-40b8-8625-f8c7d0252b2a
# ╟─a9d74a5b-c228-4389-a55e-1bbbb8ae988a
# ╟─a122646f-4f99-4f2d-bd79-e4f5fc998010
# ╟─624cbd17-40a7-4f37-bae8-558e2bb0b6dc
# ╟─d07d83de-0473-4f55-93cb-a5d976882418
# ╟─9de3cfdf-c499-4d64-bc4b-6efa98537fb3
# ╟─e7f4bb7f-1f56-4b1a-9609-2f2e60c49ce8
# ╟─027fc87f-652c-41e8-ad77-84d8a223fedd
# ╟─3eec3ecc-b526-4010-a4f8-f6ee22a6c2c2
# ╟─7314ed82-0871-493b-8af2-11be4e0b8077
# ╟─4898cf5b-50d2-4dc6-9c3d-6195c5ba21a3
# ╟─a72ad4c1-1c89-475e-9638-0188bd19a006
# ╟─1cf38e77-ce5e-457e-a8e9-cd44a544011b
# ╟─134ea9f0-bd47-4d23-8dac-f5ac4c00c02c
# ╟─afc8d58c-65c8-4945-b52d-b99c6e91c719
# ╟─988ecb55-6d61-4502-960a-afec6fdf1b36
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002