{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Modelling the recovery process using probability"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Recall code from last time for $I_{n+1} = (1 + c) I_n = \\lambda * I_n$:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"run_infection (generic function with 3 methods)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function run_infection(I_0, λ, T=20) # T=20 is default value\n",
"\n",
" I = zeros(T)\n",
" I[1] = I_0\n",
"\n",
" for n in 1:T-1\n",
" I[n+1] = λ * I[n] \n",
" end\n",
"\n",
" return I\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"run_infection(1.0, 0.1);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Adding new data to extend a vector"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1-element Array{Float64,1}:\n",
" 1.0"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v = [1.0]"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Array{Float64,1}:\n",
" 1.0\n",
" 7.0"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"push!(v, 7.0) # ! means: function `push!` modifies its first argument"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"length(v)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"ename": "BoundsError",
"evalue": "BoundsError: attempt to access 2-element Array{Float64,1} at index [3]",
"output_type": "error",
"traceback": [
"BoundsError: attempt to access 2-element Array{Float64,1} at index [3]",
"",
"Stacktrace:",
" [1] getindex(::Array{Float64,1}, ::Int64) at ./array.jl:787",
" [2] top-level scope at In[14]:2"
]
}
],
"source": [
"a = 1\n",
"\n",
"v[3]"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Array{Float64,1}:\n",
" 1.0\n",
" 7.0"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Float64,1}:\n",
" 1.0\n",
" 7.0\n",
" 10.0"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[v; 10]"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2-element Array{Float64,1}:\n",
" 1.0\n",
" 7.0"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Float64,1}:\n",
" 1.0\n",
" 7.0\n",
" 10.0"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v2 = [v; 10]"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Float64,1}:\n",
" 1.0\n",
" 7.0\n",
" 10.0"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v2 = copy(v)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"run_infection (generic function with 3 methods)"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function run_infection(I_0, λ, T=20) # T=20 is default value\n",
"\n",
" Is = [I_0]\n",
" \n",
" I = I_0 # current value of I\n",
"\n",
" for n in 1:T-1\n",
" I_next = λ * I\n",
" \n",
" push!(Is, I_next)\n",
" end\n",
"\n",
" return Is\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"20-element Array{Float64,1}:\n",
" 1.0\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1\n",
" 1.1"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"run_infection(1.0, 1.1)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"run_infection (generic function with 3 methods)"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function run_infection(I_0, λ, T=20) # T=20 is default value\n",
"\n",
" Is = [I_0]\n",
" I = I_0 # current value of I\n",
"\n",
" for n in 1:T-1\n",
" I_next = λ * I\n",
" \n",
" push!(Is, I_next)\n",
" \n",
" I = I_next\n",
" end\n",
" \n",
" return Is\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"20-element Array{Float64,1}:\n",
" 1.0\n",
" 1.1\n",
" 1.2100000000000002\n",
" 1.3310000000000004\n",
" 1.4641000000000006\n",
" 1.6105100000000008\n",
" 1.771561000000001\n",
" 1.9487171000000014\n",
" 2.1435888100000016\n",
" 2.357947691000002\n",
" 2.5937424601000023\n",
" 2.853116706110003\n",
" 3.1384283767210035\n",
" 3.4522712143931042\n",
" 3.797498335832415\n",
" 4.177248169415656\n",
" 4.594972986357222\n",
" 5.054470284992944\n",
" 5.559917313492239\n",
" 6.115909044841463"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"run_infection(1.0, 1.1)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3-element Array{Float64,1}:\n",
" 1.0\n",
" 7.0\n",
" 10.0"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4-element Array{Float64,1}:\n",
" 8.0\n",
" 1.0\n",
" 7.0\n",
" 10.0"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pushfirst!(v, 8) # deque - double-ended queue"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Randomness"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.6254465754499758"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r = rand()"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.9251027927183173"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r = rand()"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [],
"source": [
"using Random"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [],
"source": [
"Random.seed!(3);"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.8116984049958615"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r = rand()"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.9884323655013432"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r = rand()"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [],
"source": [
"Random.seed!(3); # mechanism to get repeatable sequences of random numbers"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.8116984049958615"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r = rand()"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.9884323655013432"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r = rand()"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10-element Array{Float64,1}:\n",
" 0.8076220876500786\n",
" 0.9700908450487538\n",
" 0.14006111319509862\n",
" 0.5094438024440222\n",
" 0.05869740597593154\n",
" 0.004257960600515309\n",
" 0.9746379934512355\n",
" 0.5572251384524507\n",
" 0.4644219211852372\n",
" 0.2875090558291695"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rand(10)"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [],
"source": [
"using Plots"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [],
"source": [
"r = rand(50);"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"scatter(r)"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"one(0.5)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"scatter(r, 0.5 .* one.(r), ylim=(0, 1))"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
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"type": "node"
},
"text/html": [
"\n",
" \n",
"\n"
],
"text/plain": [
"Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Scope(Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :label), Any[\"n\"], Dict{Symbol,Any}(:className => \"interact \",:style => Dict{Any,Any}(:padding => \"5px 10px 0px 10px\")))], Dict{Symbol,Any}(:className => \"interact-flex-row-left\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :input), Any[], Dict{Symbol,Any}(:max => 100,:min => 1,:attributes => Dict{Any,Any}(:type => \"range\",Symbol(\"data-bind\") => \"numericValue: index, valueUpdate: 'input', event: {change: function (){this.changes(this.changes()+1)}}\",\"orient\" => \"horizontal\"),:step => 1,:className => \"slider slider is-fullwidth\",:style => Dict{Any,Any}()))], Dict{Symbol,Any}(:className => \"interact-flex-row-center\")), Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{WebIO.DOM}(WebIO.DOM(:html, :p), Any[], Dict{Symbol,Any}(:attributes => Dict(\"data-bind\" => \"text: formatted_val\")))], Dict{Symbol,Any}(:className => \"interact-flex-row-right\"))], Dict{Symbol,Any}(:className => \"interact-flex-row interact-widget\")), Dict{String,Tuple{Observables.AbstractObservable,Union{Nothing, Bool}}}(\"changes\" => (Observable{Int64} with 1 listeners. Value:\n",
"0, nothing),\"index\" => (Observable{Int64} with 2 listeners. Value:\n",
"50, nothing)), Set(String[]), nothing, Asset[Asset(\"js\", \"knockout\", \"/Users/dpsanders/.julia/packages/Knockout/IP1uR/src/../assets/knockout.js\"), Asset(\"js\", \"knockout_punches\", \"/Users/dpsanders/.julia/packages/Knockout/IP1uR/src/../assets/knockout_punches.js\"), Asset(\"js\", nothing, \"/Users/dpsanders/.julia/packages/InteractBase/9mFwe/src/../assets/all.js\"), Asset(\"css\", nothing, \"/Users/dpsanders/.julia/packages/InteractBase/9mFwe/src/../assets/style.css\"), Asset(\"css\", nothing, \"/Users/dpsanders/.julia/packages/Interact/cdOnS/src/../assets/bulma_confined.min.css\")], Dict{Any,Any}(\"changes\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")],\"index\" => Any[WebIO.JSString(\"(function (val){return (val!=this.model[\\\"index\\\"]()) ? (this.valueFromJulia[\\\"index\\\"]=true, this.model[\\\"index\\\"](val)) : undefined})\")]), WebIO.ConnectionPool(Channel{Any}(sz_max:32,sz_curr:0), Set(AbstractConnection[]), Base.GenericCondition{Base.AlwaysLockedST}(Base.InvasiveLinkedList{Task}(Task (runnable) @0x00000001342ec6d0, Task (runnable) @0x00000001342ec6d0), Base.AlwaysLockedST(1))), WebIO.JSString[WebIO.JSString(\"function () {\\n var handler = (function (ko, koPunches) {\\n ko.punches.enableAll();\\n ko.bindingHandlers.numericValue = {\\n init: function(element, valueAccessor, allBindings, data, context) {\\n var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n stringified.subscribe(function(value) {\\n var val = parseFloat(value);\\n if (!isNaN(val)) {\\n valueAccessor()(val);\\n }\\n });\\n valueAccessor().subscribe(function(value) {\\n var str = JSON.stringify(value);\\n if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n return;\\n if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n return;\\n stringified(str);\\n });\\n ko.applyBindingsToNode(\\n element,\\n {\\n value: stringified,\\n valueUpdate: allBindings.get('valueUpdate'),\\n },\\n context,\\n );\\n }\\n };\\n var json_data = {\\\"formatted_vals\\\":[\\\"1\\\",\\\"2\\\",\\\"3\\\",\\\"4\\\",\\\"5\\\",\\\"6\\\",\\\"7\\\",\\\"8\\\",\\\"9\\\",\\\"10\\\",\\\"11\\\",\\\"12\\\",\\\"13\\\",\\\"14\\\",\\\"15\\\",\\\"16\\\",\\\"17\\\",\\\"18\\\",\\\"19\\\",\\\"20\\\",\\\"21\\\",\\\"22\\\",\\\"23\\\",\\\"24\\\",\\\"25\\\",\\\"26\\\",\\\"27\\\",\\\"28\\\",\\\"29\\\",\\\"30\\\",\\\"31\\\",\\\"32\\\",\\\"33\\\",\\\"34\\\",\\\"35\\\",\\\"36\\\",\\\"37\\\",\\\"38\\\",\\\"39\\\",\\\"40\\\",\\\"41\\\",\\\"42\\\",\\\"43\\\",\\\"44\\\",\\\"45\\\",\\\"46\\\",\\\"47\\\",\\\"48\\\",\\\"49\\\",\\\"50\\\",\\\"51\\\",\\\"52\\\",\\\"53\\\",\\\"54\\\",\\\"55\\\",\\\"56\\\",\\\"57\\\",\\\"58\\\",\\\"59\\\",\\\"60\\\",\\\"61\\\",\\\"62\\\",\\\"63\\\",\\\"64\\\",\\\"65\\\",\\\"66\\\",\\\"67\\\",\\\"68\\\",\\\"69\\\",\\\"70\\\",\\\"71\\\",\\\"72\\\",\\\"73\\\",\\\"74\\\",\\\"75\\\",\\\"76\\\",\\\"77\\\",\\\"78\\\",\\\"79\\\",\\\"80\\\",\\\"81\\\",\\\"82\\\",\\\"83\\\",\\\"84\\\",\\\"85\\\",\\\"86\\\",\\\"87\\\",\\\"88\\\",\\\"89\\\",\\\"90\\\",\\\"91\\\",\\\"92\\\",\\\"93\\\",\\\"94\\\",\\\"95\\\",\\\"96\\\",\\\"97\\\",\\\"98\\\",\\\"99\\\",\\\"100\\\"],\\\"changes\\\":WebIO.getval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"2580909790707749940\\\",\\\"id\\\":\\\"ob_10\\\",\\\"type\\\":\\\"observable\\\"}),\\\"index\\\":WebIO.getval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"2580909790707749940\\\",\\\"id\\\":\\\"ob_09\\\",\\\"type\\\":\\\"observable\\\"})};\\n var self = this;\\n function AppViewModel() {\\n for (var key in json_data) {\\n var el = json_data[key];\\n this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n }\\n \\n [this[\\\"formatted_val\\\"]=ko.computed( function(){\\n return this.formatted_vals()[parseInt(this.index())-(1)];\\n }\\n,this)]\\n [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"2580909790707749940\\\",\\\"id\\\":\\\"ob_10\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"2580909790707749940\\\",\\\"id\\\":\\\"ob_09\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n \\n }\\n self.model = new AppViewModel();\\n self.valueFromJulia = {};\\n for (var key in json_data) {\\n self.valueFromJulia[key] = false;\\n }\\n ko.applyBindings(self.model, self.dom);\\n}\\n);\\n (WebIO.importBlock({\\\"data\\\":[{\\\"name\\\":\\\"knockout\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/800fa077652c804a9c5f68b6c1e29c6c61f85df7-knockout.js\\\"},{\\\"name\\\":\\\"knockout_punches\\\",\\\"type\\\":\\\"js\\\",\\\"url\\\":\\\"/assetserver/81e3f253b58efa296c3c65b00e2bf7b25de6cca9-knockout_punches.js\\\"}],\\\"type\\\":\\\"async_block\\\"})).then((imports) => handler.apply(this, imports));\\n}\\n\")])], Dict{Symbol,Any}(:className => \"field interact-widget\")), Observable{Any} with 0 listeners. Value:\n",
"Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Plot{Plots.GRBackend() n=1}], Dict{Symbol,Any}(:className => \"interact-flex-row interact-widget\"))], Dict{Symbol,Any}())"
]
},
"execution_count": 52,
"metadata": {
"application/vnd.webio.node+json": {
"kernelId": "892ce113-1b6c-4df8-899f-be085cea6efe"
}
},
"output_type": "execute_result"
}
],
"source": [
"using Interact\n",
"\n",
"num_points = 100\n",
"r = rand(num_points)\n",
"\n",
"@manipulate for n in 1:num_points\n",
" scatter(r[1:n], 0.5 .* one.(r[1:n]), ylim=(0, 1), xlim=(0, 1))\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Event with probability $p$"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"false"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p = 0.25\n",
"r = rand()\n",
"\n",
"if r < p\n",
" true\n",
"else\n",
" false\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\"Bernoulli trial\""
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"bernoulli (generic function with 1 method)"
]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function bernoulli(p)\n",
"\n",
" r = rand()\n",
"\n",
" if r < p\n",
" return true\n",
" else\n",
" return false\n",
" end\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"false"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r < p ? true : false # ternary operator"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"false"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r < p"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"bernoulli (generic function with 1 method)"
]
},
"execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function bernoulli(p)\n",
"\n",
" r = rand()\n",
"\n",
" return r < p\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {},
"outputs": [],
"source": [
"p = 0.25\n",
"trials = [bernoulli(p) for i in 1:100];\n"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n"
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"scatter(trials)"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"25"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"count(trials)"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"29"
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"trials = [bernoulli(p) for i in 1:100];\n",
"count(trials)"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"30"
]
},
"execution_count": 99,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"trials = [bernoulli(p) for i in 1:100];\n",
"count(trials)"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"bernoulli_experiment (generic function with 2 methods)"
]
},
"execution_count": 106,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function bernoulli_experiment(p, N=100)\n",
" trials = [bernoulli(p) for i in 1:N];\n",
" return count(trials)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"100"
]
},
"execution_count": 103,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"count(trials .== false) + count(trials)"
]
},
{
"cell_type": "code",
"execution_count": 104,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"70"
]
},
"execution_count": 104,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"count(.!(trials)) # ! is not"
]
},
{
"cell_type": "code",
"execution_count": 107,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"29"
]
},
"execution_count": 107,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bernoulli_experiment(0.25)"
]
},
{
"cell_type": "code",
"execution_count": 108,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"33"
]
},
"execution_count": 108,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bernoulli_experiment(0.25)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A function which has different **outcomes** in different runs is called a **random variable**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Monte Carlo simulation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Run the same random process a lot of times and look at the results"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"100-element Array{Int64,1}:\n",
" 3\n",
" 5\n",
" 7\n",
" 6\n",
" 4\n",
" 3\n",
" 4\n",
" 4\n",
" 6\n",
" 5\n",
" 7\n",
" 1\n",
" 3\n",
" ⋮\n",
" 7\n",
" 7\n",
" 8\n",
" 5\n",
" 5\n",
" 4\n",
" 2\n",
" 5\n",
" 6\n",
" 4\n",
" 4\n",
" 4"
]
},
"execution_count": 113,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p = 0.25\n",
"N = 20 # num of trials\n",
"num_expts = 100 \n",
"\n",
"results = [bernoulli_experiment(p, N) for i in 1:num_expts]"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"