### A Pluto.jl notebook ###
# v0.18.1

using Markdown
using InteractiveUtils

# ╔═╡ 521890de-ab23-11ec-0c2f-2dcaee6dc1bc
using Plots, Distributions, PlutoUI

# ╔═╡ dbf8cdd6-7815-49f8-9dd6-5987000792ce
md"""
# 3월24일 강의영상
"""

# ╔═╡ 20697867-c74a-4485-bb97-c6a31060669a
html"""
<div notthestyle="position: relative; right: 0; top: 0; z-index: 300;"><iframe src="https://www.youtube.com/embed/playlist?list=PLQqh36zP38-xIz4hvPmHZz0g8g04MJ0xZ" width=600 height=375  frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe></div>
"""

# ╔═╡ 64e29d20-aaf5-4fa0-ad03-b283dac52dce
md"""
# 3월24일 강의노트
"""

# ╔═╡ 3f43a098-0653-4cb3-a48e-673e342ae48b
Plots.plotly()

# ╔═╡ ddae88bf-0b17-40f4-ae33-5cd70aa8a0de
md"""
### 포아송분포 ($X \sim Poi(\lambda)$)
"""

# ╔═╡ c0e571b7-580a-4b25-ba5b-6cb729ec36b9
md"""
`-` 포아송분포의 요약 
- X의의미: 발생횟수의 평균이 λ인 분포에서 실제 발생횟수를 X라고 한다. 
- X의범위: 발생안할수도 있으므로 X=0이 가능. 따라서 X=0,1,2,3,... 
- 파라메터의 의미와 범위: λ = 평균적인 발생횟수; λ>0. 
- pdf: 
- mgf: 
- E(X): λ
- V(X): λ
"""

# ╔═╡ 7807af0d-012a-432a-9a64-5df4a633333b
md"""
`-` [포아송분포의 예시](https://www.statology.org/poisson-distribution-real-life-examples/#:~:text=Example%201%3A%20Calls%20per%20Hour,receives%2010%20calls%20per%20hour.)
- Calls per Hour at a Call Center
- Number of Arrivals at a Restaurant
- Number of Website Visitors per Hour
- Number of Bankruptcies Filed per Month
- Number of Network Failures per Week
"""

# ╔═╡ 713fc6aa-3cdb-49ad-903c-9127452e69b9
md"""
`-` 평균이 3인 포아송분포에서 100개의 샘플을 뽑는 방법 
"""

# ╔═╡ 443fc984-8109-45c3-a6e5-e6e565dd0491
md"""
(방법1)
"""

# ╔═╡ 6d30adc5-4da9-4e85-bd47-e128de6fffb2
rand(Poisson(3),100)

# ╔═╡ 9a9ed77b-903e-4587-967c-518343a62e54
md"""
(방법2) 이항분포의 포아송근사를 이용 
- 이론: 이항분포에서 (1) $n\to \infty$ (2) $p\to 0$ (3) $np=\lambda$ 이면 이것은 평균이 $\lambda$인 포아송분포로 근사함. 
- 평균이 $\lambda$인 포아송분포는 $B(n,\frac{\lambda}{n})$로 근사할 수 있다. 이때 $n$이 커질수록 더 정확해짐. 
"""

# ╔═╡ b0ca1c8f-272d-4a2f-ac4b-10bd659fceda
let 
	samplesize_of_poisson = 10000
	λ=3 
	n=10000
	p=λ/n
	X = rand(Binomial(n,p),samplesize_of_poisson)
	Y = rand(Poisson(λ),samplesize_of_poisson)
	p1= histogram(X)
	p2= histogram(Y)
	plot(p1,p2,layout=(2,1))
end 

# ╔═╡ 1e943933-7266-4696-9388-6060d0d173e8
md"""
-  $n=10000$ 정도이면 꽤 비슷함. 
- 방법2는 근사방법이므로 엄밀히 말하면 분포를 뽑는 기법이라고 볼 수는 없음. 
"""

# ╔═╡ 4520bf81-0491-4a47-99c4-45680964654c
let 
	λ=3
	n=600
	p=λ/n
	Δt = (60/n) # 단위가 60초니까 60
	
	poi_samplesize = 10000
	X = [(rand(n) .< p) |> sum for k in 1:poi_samplesize]
	p1= histogram(X)
	p2= rand(Poisson(λ),poi_samplesize) |> histogram
	_p = plot(p1,p2,layout=(2,1))
	

	md"""
	(방법3) 균등분포 -> 베르누이 -> 이항분포 -> 포아송근사  
	- 1분동안 맥도날드에 평균 3명이 온다고 생각하자. 
	- 이건 사실 1초에 성공확률이 0.05인 베르누이 시행을 1번 시행하여 1분동안 총 60회 반복한 것으로 이해할 수 있음. 
	- 아니야, 이건 사실 $(Δt) 초에 성공확률이 $p 인 베르누이 시행을 1번 시행하여 1분동안 총 $n 회 반복한 것으로 이해할 수 있음. 
	- 아니야 아냐야, 이건 사실.... (무한반복)
	- 느낌: 하여튼 엄청 작은 시간에 엄청 작은 확률의 베르누이 시행이 독립적으로 엄청 많이 반복되는 느낌을 기억하세요!
	$(_p)
	- 위: 유니폼 -> 베르누이 -> 이항분포 -> 포아송 
	- 아래: 포아송
	"""
end 

# ╔═╡ ea7e4bb9-5eca-4046-8209-51489614c636
md"""
(방법4) 균등분포 -> inverse cdf method를 이용해서 생성할 수 있음. 
"""

# ╔═╡ 8352dd50-1f7d-4e3d-9365-5af24d43547c
md"""
`-` 포아송분포의 합은 다시 포아송분포가 된다. 
- 이론: $X \sim Poi(\lambda_1), Y\sim Poi(\lambda_2),~ X \perp Y \Rightarrow X + Y \sim P(\lambda_1 + \lambda_2)$ 
- 의미? (1) 1분동안 맥도날드 매장에 들어오는 남자의 수는 평균이 5인 포아송 분포를 따름 (2) 1분동안 맥도날드 매장에 들어오는 여자의 수는 평균이 4.5인 포아송분포를 따름 (3) 남자와 여자가 매장에 오는 사건은 독립 => 1분동안 맥도날드 매장에 오는 사람은 평균이 9.5인 포아송 분포를 따른다는 의미. 
"""

# ╔═╡ af1c928e-ab6c-4212-88df-e3109503a157
md"""
(실습)
"""

# ╔═╡ 5d376188-fad1-48fb-86a2-7dcc908c337f
let 
	n= 1000
	X = rand(Poisson(5),n)
	Y = rand(Poisson(4.5),n)
	p1 = X.+Y |> histogram 
	p2 = rand(Poisson(9.5),n) |> histogram
	plot(p1,p2,layout=(2,1))
end

# ╔═╡ 0ab7a2d2-0d17-4523-8e91-b28624303524
let
	n=1000
	λ=5 
	X = rand(Poisson(λ),n) 
	md"""
	`-` 평균과 분산의 추정 
	- 평균: $λ
	- 평균의 추정치: $(mean(X))
	- 분산: $λ
	- 분산의 추정치: $(var(X))
	"""
end

# ╔═╡ f140fb8e-308f-40e4-a199-b0e0d20dc7da
md"""
- 생각해보니까 왜 평균추정값과 분산추정값이 달라야하나? 
"""

# ╔═╡ a222e7a1-dd18-4416-92eb-320f926d1d77
let 
	n = 10000
	λ = 5 
	p1=[mean(rand(Poisson(λ),n)) for k in 1:100] |> histogram
	p2=[var(rand(Poisson(λ),n)) for k in 1:100] |> histogram
	#p3=[(mean(rand(Poisson(λ),n))+var(rand(Poisson(λ),n)))/2  for k in 1:100] |> histogram
	_p=plot(p1,p2,layout=(2,1))
	md""" 
	mean(X),var(X)로 λ를 추정
	$(_p)
	- 위: mean(X)
	- 아래: var(X)
	"""
end 

# ╔═╡ ebe47018-7a6d-42a4-a61a-0079b5c077f8
md"""
- 히스토그램을 그려보니까 누가봐도 mean(X)로 λ를 추정하는것이 var(X)로 λ를 추정하는것 보다 좋아 보인다. 
- 그냥 mean(X)=평균추정량=분산추정량 이라고 주장하면 안되나? => 됩니다! 이게 바로 MLE에요! 
"""

# ╔═╡ 29a2e115-37a7-4f67-a040-a4c1648fdfa6
md"""
### 지수분포 ($X \sim Exp(1/\lambda)$)
"""

# ╔═╡ a185e773-f51e-449b-8d40-de819bd4badf
md"""
`-` 지수분포의 요약 
- X의의미: 시간1에 평균적으로 λ번 발생하는 시건이 있을때 첫 번째 이벤트가 발생할때 까지 걸리는 시간. 
- X의범위: 시간은 양수이므로 X ≥ 0
- 파라메터의 의미와 범위: (1) λ = 시간1에 평균적으로 발생하는 이벤트의 수 (2) 1/λ = 한번의 이벤트가 발생할때까지 평균적으로 걸리는 시간; λ>0 
- pdf: $f(x)=\lambda e^{-\lambda x}$ 
- mgf: 
- cdf: $F(x)=1-e^{-\lambda x}$
- E(X): $\frac{1}{\lambda}$
- V(S): $\frac{1}{\lambda^2}$
"""

# ╔═╡ 885941da-8604-4af0-8b69-6d564348f116
md"""
`-` 평균이 10인 지수분포에서 100개의 샘플을 뽑는 방법 
"""

# ╔═╡ 2134273b-fca8-4238-b906-f05948fae7d2
md"""
(방법1)
"""

# ╔═╡ 1da1e41d-73a6-43f4-9d69-3e28b83da8e0
rand(Exponential(10),100)

# ╔═╡ caa4ea34-e2ae-455a-88a7-6324ff2cc8ef
md"""
(방법2) 포아송프로세스 -> 지수분포
- 맥도날드에 시간1당 0.1명씩 평균적으로 방문한다. 1명 방문하는데에는 평균적으로 시간이 10이 걸린다고 볼 수 있음. 
- 따라서 언뜻생각하면 포아송과 지수분포는 역의 관계라서 포아송분포를 만들고 역수를 취하면 지수분포를 쉽게 만들 수 있을 것 같다.
"""

# ╔═╡ eb7572bd-63eb-4792-b982-9f320358a99c
rand(Poisson(0.1),100) # 0이 나오네?

# ╔═╡ 1c902aac-d633-4cde-95e1-62e42e0ffe87
md"""
- 0이 나온다?
- 생각해보니까 0이 없다고 쳐도 나올 수 있는 값은 1, 1/2, 1/3, 1/4, ... 따위임 (애초에 틀린 접근)
- 아이디어: 극한의 베르누이로 포아송을 만들때, 몇번 성공했는지 관심을 가지고 카운팅 했음 => 조금 응용해서 첫 성공까지 몇번의 시도를 해야하는지 카운팅을 한다고 생각하면 시간계산이 가능할것 같다. 
- 결국 포아송프로세스 -> 지수분포로 가야함
"""

# ╔═╡ 09a0f94b-e90c-4233-933c-7c1237f11658
md"""
(예비학습)
"""

# ╔═╡ 8a1a0dd9-a5fc-4ea1-b026-c47490db1bc3
rand() # 유니폼에서 1개의 샘플 추출 

# ╔═╡ 011a3508-939b-47b6-a062-9d61c6ea9eec
let 
	i=0 
	while i <= 5
		i=i+1
	end
	i
end

# ╔═╡ aef02ab9-4e12-4030-b428-f4aa36f65ba6
md"""
(풀이시작)
"""

# ╔═╡ e5ac4d16-20ed-4c35-832f-b6fd5cfad2dd
function try_until_you_succeed(p) # 성공할때까지 시도하는 함수 
	n_of_try=0 
	u=0
	while u < (1-p) # p=0이면 무한반복 
		u = rand()
		n_of_try = n_of_try + 1 
	end
	return n_of_try 
end

# ╔═╡ b0d76b5d-cefa-489e-9963-830d6fb710a0
[try_until_you_succeed(0.1) for k in 1:10000] |> histogram

# ╔═╡ a27efaae-2793-4d8a-8dd6-4900caa80a4a
let 
	exp_samplesize = 10000 
	λ=0.1
	n=10000
	p=λ/n
	Δt = (1/n) # 단위가 시간1이니까 
	X = [try_until_you_succeed(p) for k in 1:exp_samplesize] .* Δt
	p1 = X|> histogram
	p2 = rand(Exponential(10),exp_samplesize) |> histogram
	plot(p1,p2,layout=(2,1))
end 

# ╔═╡ 829ad68b-7352-4d18-bcfd-90919b18648c
md"""
- 불평: 샘플하나뽑는데 시간이 오래걸림. (정확도를 올릴수록 더 오래걸림)
"""

# ╔═╡ fc1acb74-f856-47c7-b9e1-8d635f0e35ca
md"""
(방법3) inverse cdf method 
- 이론적인 pdf를 알고 있다는 전제가 필요함. 
- 자세하게 살펴보자. 
"""

# ╔═╡ c2c198a7-4597-49e2-98d7-8776603db7fa
md"""
##### Inverse cdf method를 활용하여 지수분포에서 샘플추출 
"""

# ╔═╡ cf6a955e-d684-4a16-bc0f-14409f83536a
md"""
`-` 아래와 같은 2개의 지수분포의 pdf를 고려하자.

$$f(x)=e^{-x}$$ 

$$g(x)=\frac{1}{5}e^{-\frac{1}{5}x}$$
"""

# ╔═╡ ed777a37-6351-41a1-aed3-86412c9edaac
md"""
`-` 각각의 pdf를 그려보면 아래와 같다. 
"""

# ╔═╡ db26ed1c-b01a-4f06-886d-4079cf2c139f
let 
	p1= plot(x-> exp(-x), 0, 20)
	p2= plot(x-> 1/5* exp(-x/5),0,20)
	plot(p1,p2,layout=(2,1))
end 

# ╔═╡ 7889eeaa-a7b5-4cdd-8b26-c0de2d2981e0
md"""
`-` 이번에는 각각의 cdf를 그려보자. 

$$F(x)=\int_0^x f(\tau)d\tau=\int_0^x e^{-\tau} d\tau = \left[-e^{-\tau}\right]_0^x=1-e^{-x}$$

$$G(x)=\int_0^x g(\tau)d\tau=\int_0^x \frac{1}{5}e^{-\tau/5} d\tau = \left[-e^{-\tau/5}\right]_0^x=1-e^{-x/5}$$

"""

# ╔═╡ 23abf65d-276d-4a5d-ae70-b41255b35b48
let 
	p1= plot(x -> 1-exp(-x), 0, 20)
	p2= plot(x -> 1-exp(-x/5), 0, 20)
	plot(p1,p2,layout=(2,1))
end 

# ╔═╡ ac1c2222-04b5-4af0-a55c-723b1ad57dec
md"""
`-` cdf 해석 
- 위(평균이1인지수분포) = 5정도면 거의 cdf의 값이 1에 가까워짐 
- 아래(평균이5인지수분포) = 5정도에서 값이 거의 0.63정도임 => 100번뽑으면 5보다 작은게 63개정도.. 

`-` cdf의 y축에서 랜덤변수를 발생시킨다음 $\rightarrow \downarrow$ 와 같이 이동하여 $x$축에 내린다고 생각해보자. 
- 위: 대부분 5이하에 떨어짐 
- 아래: 약 63% 정도만 5이하에 떨어짐.
"""

# ╔═╡ 9c2f4080-9dd5-4f85-83f7-42bf3c719e6e
# F(x) = 1-exp(-x)
# F: x-> *(-1) -> exp -> *(-1) -> +1 
# Finv: y-> -1 -> *(-1) -> log -> *(-1) 
Finv(x) = -log(-(x-1))
 Ginv(x) =-5*log(-(x-1))

# ╔═╡ 23118be8-96de-4df1-8b01-297f6ac3d1be
 Ginv(x) =-5*log(-(x-1))

# ╔═╡ 7520499e-e499-4873-b96a-acbdb777ca80


# ╔═╡ 2a78805b-6f73-4a44-b862-8fa15be1a20a


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[[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"
"""

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