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"source": [
"# Standard data science\n",
"import pandas as pd\n",
"import numpy as np\n",
"\n",
"%load_ext autoreload\n",
"%autoreload 2\n",
"\n",
"# Options for pandas\n",
"pd.options.display.max_columns = 20\n",
"\n",
"# Display all cell outputs\n",
"from IPython.core.interactiveshell import InteractiveShell\n",
"InteractiveShell.ast_node_interactivity = 'all'\n",
"\n",
"# Visualizations\n",
"import plotly.plotly as py\n",
"import plotly.graph_objs as go\n",
"from plotly.offline import iplot\n",
"\n",
"# Cufflinks for dataframes\n",
"import cufflinks as cf\n",
"cf.go_offline()\n",
"cf.set_config_file(world_readable=True, theme='white')"
]
},
{
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"execution_count": 164,
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"end_time": "2019-01-19T16:20:31.616672Z",
"start_time": "2019-01-19T16:20:31.566413Z"
}
},
"outputs": [],
"source": [
"from scipy.special import factorial\n",
"import pymc3 as pm"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Distribution"
]
},
{
"cell_type": "code",
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"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T16:22:12.424622Z",
"start_time": "2019-01-19T16:22:12.372661Z"
}
},
"outputs": [],
"source": [
"spins = 100\n",
"rate = 1/38\n",
"\n",
"def calc_prob_number(rate, number, trials):\n",
" lam = rate * trials\n",
" return np.exp(-lam) * (np.power(lam, number)) / factorial(number)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Expected successes per 1000 spins"
]
},
{
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"source": [
"lam = rate*spins\n",
"lam"
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"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T16:22:20.016643Z",
"start_time": "2019-01-19T16:22:19.964511Z"
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},
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"source": [
"ns = np.arange(100)\n",
"p_n = calc_prob_number(rate, ns, spins)"
]
},
{
"cell_type": "code",
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"metadata": {
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"outputs": [],
"source": [
"ra"
]
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"source": [
"Alternative using numpy"
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"\n",
"df = pd.DataFrame({'n': n})\n",
"df['n'].iplot(kind='hist', xTitle='successes / 1000 spins', \n",
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{
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{
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},
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{
"cell_type": "code",
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{
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{
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"source": [
"Probability of waiting more than 100 spins."
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{
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"end_time": "2019-01-19T15:48:14.881123Z",
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{
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"end_time": "2019-01-19T15:49:04.103724Z",
"start_time": "2019-01-19T15:49:04.047049Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0.062176524022116375\n",
"0.9378234759778836\n",
"0.23384835517828695\n"
]
}
],
"source": [
"from math import pow, exp, factorial\n",
"\n",
"class Exponential:\n",
"\n",
" def __init__(self, rate):\n",
" self.rate = rate\n",
"\n",
" def prob_less_than_or_equal(self, t):\n",
" rate = self.rate * t\n",
" return 1 - exp(-rate)\n",
"\n",
" def prob_greater_than(self, t):\n",
" return 1 - self.prob_less_than_or_equal(t)\n",
"\n",
" def prob_between(self, t1, t2):\n",
" p1 = self.prob_less_than_or_equal(t1)\n",
" p2 = self.prob_less_than_or_equal(t2)\n",
"\n",
" return p2 - p1\n",
"\n",
"expo = Exponential(1/38)\n",
"print(expo.prob_greater_than(100))\n",
"print(expo.prob_less_than_or_equal(100))\n",
"print(expo.prob_between(50, 150))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Arrivals"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T15:49:50.778559Z",
"start_time": "2019-01-19T15:49:50.725894Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"array([0.63641041, 0.31435598])"
]
},
"execution_count": 102,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rand.rand(2)"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T15:53:29.849482Z",
"start_time": "2019-01-19T15:53:29.797995Z"
}
},
"outputs": [],
"source": [
"spins = 10000\n",
"time_between = 36 \n",
"\n",
"rand = np.random.RandomState(42)\n",
"success_times = spins * time_between * np.sort(rand.rand(spins))"
]
},
{
"cell_type": "code",
"execution_count": 115,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T15:53:49.630283Z",
"start_time": "2019-01-19T15:53:49.358332Z"
}
},
"outputs": [
{
"data": {
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"config": {
"linkText": "Export to plot.ly",
"plotlyServerURL": "https://plot.ly",
"showLink": true
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"histfunc": "count",
"histnorm": "",
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"legend": {
"bgcolor": "#FFFFFF",
"font": {
"color": "#4D5663"
}
},
"paper_bgcolor": "#FFFFFF",
"plot_bgcolor": "#FFFFFF",
"title": "Histogram of Waiting Times",
"titlefont": {
"color": "#4D5663"
},
"xaxis": {
"gridcolor": "#E1E5ED",
"showgrid": true,
"tickfont": {
"color": "#4D5663"
},
"title": "Waiting Time",
"titlefont": {
"color": "#4D5663"
},
"zerolinecolor": "#E1E5ED"
},
"yaxis": {
"gridcolor": "#E1E5ED",
"showgrid": true,
"tickfont": {
"color": "#4D5663"
},
"title": "Count",
"titlefont": {
"color": "#4D5663"
},
"zerolinecolor": "#E1E5ED"
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},
"text/html": [
""
],
"text/vnd.plotly.v1+html": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"waiting_times = np.diff(success_times)\n",
"\n",
"df = pd.DataFrame({'waiting_time': waiting_times})\n",
"df['waiting_time'].iplot(kind='hist', xTitle='Waiting Time', \n",
" yTitle='Count', bins=(0, 400, 10),\n",
" title='Histogram of Waiting Times')"
]
},
{
"cell_type": "code",
"execution_count": 116,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T16:00:57.401471Z",
"start_time": "2019-01-19T16:00:57.349474Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"35.9930166887764"
]
},
"execution_count": 116,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.mean(waiting_times)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Visualizing Successes"
]
},
{
"cell_type": "code",
"execution_count": 158,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T16:13:28.062305Z",
"start_time": "2019-01-19T16:13:28.006456Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"array([[68, 2, 10, 10]])"
]
},
"execution_count": 158,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"spins = np.random.choice([0, 1], size = 100, replace=True, p=[1-rate, rate])\n",
"success_times = np.where(spins==1)[0]\n",
"waiting_times = np.diff(np.where(spins == 1))\n",
"waiting_times[:10]"
]
},
{
"cell_type": "code",
"execution_count": 159,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T16:13:28.509491Z",
"start_time": "2019-01-19T16:13:28.453164Z"
}
},
"outputs": [],
"source": [
"annotations = [go.layout.Annotation(x=x, y=1, text=str(x), ax=0, ay=200) for x in success_times]"
]
},
{
"cell_type": "code",
"execution_count": 160,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T16:13:28.913542Z",
"start_time": "2019-01-19T16:13:28.797791Z"
}
},
"outputs": [
{
"data": {
"application/vnd.plotly.v1+json": {
"config": {
"linkText": "Export to plot.ly",
"plotlyServerURL": "https://plot.ly",
"showLink": true
},
"data": [
{
"mode": "markers",
"type": "scatter",
"uid": "dc65682a-eac8-47a2-9aae-07649bdf7493",
"x": [
9,
77,
79,
89,
99
],
"y": [
1,
1,
1,
1,
1
]
}
],
"layout": {
"annotations": [
{
"ax": 0,
"ay": 200,
"text": "9",
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"y": 1
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{
"ax": 0,
"ay": 200,
"text": "77",
"x": 77,
"y": 1
},
{
"ax": 0,
"ay": 200,
"text": "79",
"x": 79,
"y": 1
},
{
"ax": 0,
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"text": "89",
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"ax": 0,
"ay": 200,
"text": "99",
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}
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""
],
"text/vnd.plotly.v1+html": [
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],
"source": [
"figure = go.Figure(data=[go.Scatter(x=success_times, y=np.ones(shape=len(success_times)), mode='markers')], \n",
" layout=go.Layout(annotations=annotations, yaxis=dict(range=(0, 1))))\n",
"iplot(figure)"
]
},
{
"cell_type": "code",
"execution_count": 134,
"metadata": {
"ExecuteTime": {
"end_time": "2019-01-19T16:08:39.468478Z",
"start_time": "2019-01-19T16:08:39.412153Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"[layout.Annotation({\n",
" 'text': ('[ 66 136 157 201 222 269 290 3' ... '6 866 890 951 954 970 971 980]'),\n",
" 'x': array([ 66, 136, 157, 201, 222, 269, 290, 326, 348, 457, 484, 523, 526, 552,\n",
" 571, 663, 671, 673, 687, 743, 751, 856, 866, 890, 951, 954, 970, 971,\n",
" 980]),\n",
" 'y': 1\n",
" })]"
]
},
"execution_count": 134,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"annotations"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"hide_input": false,
"kernelspec": {
"display_name": "Python 3",
"language": "python",
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},
"language_info": {
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"name": "ipython",
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},
"file_extension": ".py",
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"name": "python",
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"version": "3.6.5"
},
"toc": {
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"nav_menu": {},
"number_sections": true,
"sideBar": true,
"skip_h1_title": false,
"title_cell": "Table of Contents",
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"varInspector": {
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"kernels_config": {
"python": {
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"library": "var_list.py",
"varRefreshCmd": "print(var_dic_list())"
},
"r": {
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"delete_cmd_prefix": "rm(",
"library": "var_list.r",
"varRefreshCmd": "cat(var_dic_list()) "
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