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"%matplotlib inline\n",
"import numpy as np\n",
"from scipy.stats import binom, geom, hypergeom, nbinom, poisson\n",
"import plotly.plotly as py\n",
"import plotly.graph_objs as go\n",
"from plotly.offline import download_plotlyjs, init_notebook_mode, plot, iplot\n",
"import plotly.offline as offline"
]
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"# Binomial\n",
"\n",
"Recall that the pmf of $Y \\sim Binom(n,p)$ for $p \\in [0,1]$ is given by $f(y) = {n \\choose y} p^y (1-p)^y$, for $y = 0, 1, \\dots, n$. \n",
"\n",
"$E[Y] = n p$ and $Var[Y] = n p (1-p)$.\n",
"\n",
"## PMF as $p$ varies"
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""
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"source": [
"n = 40\n",
"x_range = np.arange(0,n+1)\n",
"\n",
"p_range = np.arange(0.0,1.1,0.1)\n",
"\n",
"data = [dict(\n",
" visible = False,\n",
" type = 'bar',\n",
" name = 'p = '+str(p),\n",
" x = x_range,\n",
" y = binom.pmf(x_range,n,p)) for p in p_range]\n",
"\n",
"steps = []\n",
"for i in range(len(data)):\n",
" step = dict(\n",
" method = 'restyle',\n",
" args = ['visible', [False] * len(p_range)],\n",
" label = str(p_range[i])\n",
" )\n",
" step['args'][1][i] = True # Toggle i'th trace to \"visible\"\n",
" steps.append(step)\n",
"\n",
"i = 5\n",
"data[i]['visible'] = True \n",
"\n",
"sliders = [ dict(\n",
" active = i,\n",
" currentvalue = {\"prefix\": \"p = \"},\n",
" pad = {\"t\": 50},\n",
" steps = steps\n",
") ]\n",
"\n",
"\n",
"layout = dict(sliders=sliders, \n",
" title='PMF Bernoulli(n,p) as p varies',\n",
" yaxis=dict(\n",
" autorange=False,\n",
" range=[0, 1]\n",
" ))\n",
"\n",
"fig = dict(data=data, layout=layout)\n",
"\n",
"offline.iplot(fig, filename='Binomial_var_p')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## PMF as $n$ varies"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
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"n_range = np.arange(1,41,1)\n",
"p_range = np.array([0.25,0.5,0.75])\n",
"\n",
"hist = [dict(\n",
" visible = False,\n",
" type = 'bar',\n",
" name = 'p = ' + str(p),\n",
" x = np.arange(0,n+1),\n",
" y = binom.pmf(np.arange(0,n+1),n,p)) for n in n_range for p in p_range]\n",
"\n",
"lines = [dict(\n",
" visible = False,\n",
" type = 'scatter',\n",
" name = 'µ for p = ' + str(p),\n",
" line = dict(dash = 'dash'),\n",
" opacity = 0.3,\n",
" x=[n*p, n*p],\n",
" y=[-10,10]) for n in n_range for p in p_range]\n",
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"data = hist + lines\n",
"\n",
"steps = []\n",
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" step = dict(\n",
" method = 'restyle',\n",
" args = ['visible', [False] * len(data)],\n",
" label = str(n_range[i])\n",
" )\n",
" step['args'][1][i*len(p_range) + 0] = True # Toggle i'th trace to \"visible\"\n",
" step['args'][1][i*len(p_range) + 1] = True\n",
" step['args'][1][i*len(p_range) + 2] = True\n",
" step['args'][1][len(p_range)*len(n_range) + i*len(p_range) + 0] = True\n",
" step['args'][1][len(p_range)*len(n_range) + i*len(p_range) + 1] = True\n",
" step['args'][1][len(p_range)*len(n_range) + i*len(p_range) + 2] = True\n",
" steps.append(step)\n",
"\n",
"i = 20\n",
"data[i*len(p_range) + 0]['visible'] = True \n",
"data[i*len(p_range) + 1]['visible'] = True \n",
"data[i*len(p_range) + 2]['visible'] = True \n",
"data[len(p_range)*len(n_range) + i*len(p_range) + 0]['visible'] = True \n",
"data[len(p_range)*len(n_range) + i*len(p_range) + 1]['visible'] = True \n",
"data[len(p_range)*len(n_range) + i*len(p_range) + 2]['visible'] = True \n",
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"sliders = []\n",
"sliders.append( dict(\n",
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"))\n",
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"layout = dict(sliders=sliders, \n",
" title='PMF Bernoulli(n,p) as n varies',\n",
" yaxis=dict(\n",
" autorange=False,\n",
" range=[0, .8]\n",
" ),\n",
" xaxis=dict(\n",
" tick0=0,\n",
" dtick=1\n",
" )\n",
" )\n",
"\n",
"fig = dict(data=data, layout=layout)\n",
"\n",
"offline.iplot(fig, filename='Binomial_var')"
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},
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"source": [
"# Geometric \n",
"\n",
"Recall that the pmf of $Y \\sim Geom(p)$ for $p \\in [0,1]$ is given by $f(y) = (1-p)^{y-1}p$, for $y = 1, 2 \\dots$. \n",
"\n",
"$E[Y] = \\frac{1}{p}$ and $Var[Y] = \\frac{1-p}{p^2}$.\n",
"\n",
"## PMF as $p$ varies"
]
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" name = 'f(y)',\n",
" x = np.arange(geom.ppf(0.01,p), geom.ppf(0.99,p)),\n",
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"# Poisson\n",
"\n",
"Recall that the pmf of $Y \\sim Poisson(\\lambda)$ for average $\\lambda > 0$ is given by $f(y) = \\frac{\\lambda^y}{y!} e^{-\\lambda}$, for $y = 0, 1, 2 \\dots$. \n",
"\n",
"$E[Y] = \\lambda$ and $Var[Y] = \\lambda$.\n",
"\n",
"## PMF as $\\lambda$ varies"
]
},
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" opacity = 0.3,\n",
" x=[np.sqrt(p), np.sqrt(p)],\n",
" y=[-10,10]) for p in p_range]\n",
"\n",
"data = []\n",
"data = hist + linesmu + linessig\n",
"\n",
"steps = []\n",
"for i in range(len(p_range)):\n",
" step = dict(\n",
" method = 'restyle',\n",
" args = ['visible', [False] * len(data)],\n",
" label = str(p_range[i])\n",
" )\n",
" step['args'][1][i] = True # Toggle i'th trace to \"visible\"\n",
" step['args'][1][len(p_range) + i] = True # Toggle i'th trace to \"visible\"\n",
" step['args'][1][2*len(p_range) + i] = True # Toggle i'th trace to \"visible\"\n",
" steps.append(step)\n",
"\n",
"i = 5\n",
"data[i]['visible'] = True \n",
"data[len(p_range) + i]['visible'] = True \n",
"data[2*len(p_range) + i]['visible'] = True \n",
"\n",
"sliders = [ dict(\n",
" active = i,\n",
" currentvalue = {\"prefix\": \"λ = \"},\n",
" pad = {\"t\": 50},\n",
" steps = steps\n",
") ]\n",
"\n",
"\n",
"layout = dict(sliders=sliders, \n",
" title='PMF Poisson(λ) as λ varies',\n",
" yaxis=dict(\n",
" autorange=False,\n",
" range=[0, 1]\n",
" ),\n",
" xaxis=dict(\n",
" range=[-1,40]\n",
" ))\n",
"\n",
"fig = dict(data=data, layout=layout)\n",
"\n",
"offline.iplot(fig, filename='Poisson_var')"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"# Hypergeometric\n",
"\n",
"Recall that the pmf of $Y \\sim hypergeometric(N,K,n)$ for the probability of success in $n$ draws without replacement from a population of size $N$ that contains $K$ successes is given by $$P(Y=k) = f(k) = \\frac{{N \\choose k}{N-K \\choose n-k}}{{N \\choose n}}\\,,$$ for $k$ in the range $\\max(0, n+K-N) \\leq k \\leq \\min(K,n)$. \n",
"\n",
"$E[Y] = n \\frac{K}{N}$ and $Var[Y] = n \\frac{K}{N} \\frac{N-K}{N} \\frac{N-n}{N-1}$. \n",
"\n",
"Note that the probability of success changes on each draw as each draw decreases the population since we are sampling without replacement. Question: How does this compare to the binomial distribution; that is, for a binomial experiment, how are we drawing the samples?\n",
"\n",
"Let $X \\sim binomial(n, p)$ for $p = K/N$. If ($N$ and $K$ are large compared to $n$) AND ($p$ is not close to $0$ or $1$) then $Y$ and $X$ have similar distributions:\n",
"$$P(Y \\leq k) \\approx P(X \\leq k)\\,,$$\n",
"since if $n$ is small compared to $N$ and $K$ the draws are \"closer\" to being iid $Bernoulli(p)$ trials. \n",
"\n",
"\n",
"## PMF as $n$ increases toward $N, K$ for three values of $p = K/N$"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
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"title": "PMF for hypergeom(N=1000, K, n) as n increases",
"xaxis": {
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"yaxis": {
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"source": [
"N = 1000\n",
"p_range = np.array([0.05,0.25, 0.5])\n",
"n_range = np.arange(25,475,25)\n",
"\n",
"hist = []\n",
"hist = [dict(\n",
" visible = False,\n",
" type = 'bar',\n",
" name = 'K/N = ' + str(p),\n",
" x = np.arange(0,int(p*N)+1),\n",
" # hypergeom.pmf(k, N, K, n)\n",
" y = hypergeom.pmf(np.arange(0,int(p*N)+1), N, int(p*N), n)) for n in n_range for p in p_range]\n",
"\n",
"blins = []\n",
"blines = [dict(\n",
" visible = False,\n",
" type = 'scatter',\n",
" name = 'p = ' + str(p),\n",
" x = np.arange(0, n+1),\n",
" y = binom.pmf(np.arange(0,n+1),n,p)) for n in n_range for p in p_range]\n",
"\n",
"data =[]\n",
"data = hist + blines # + lines\n",
"\n",
"steps = []\n",
"for i in range(len(n_range)):\n",
" step = dict(\n",
" method = 'restyle',\n",
" args = ['visible', [False] * len(data)],\n",
" label = str(n_range[i])\n",
" )\n",
" step['args'][1][i*len(p_range) + 0] = True # Toggle i'th trace to \"visible\"\n",
" step['args'][1][i*len(p_range) + 1] = True\n",
" step['args'][1][i*len(p_range) + 2] = True\n",
" step['args'][1][len(p_range)*len(n_range) + i*len(p_range) + 0] = True\n",
" step['args'][1][len(p_range)*len(n_range) + i*len(p_range) + 1] = True\n",
" step['args'][1][len(p_range)*len(n_range) + i*len(p_range) + 2] = True\n",
" steps.append(step)\n",
"\n",
"i = 0\n",
"data[i*len(p_range) + 0]['visible'] = True \n",
"data[i*len(p_range) + 1]['visible'] = True \n",
"data[i*len(p_range) + 2]['visible'] = True \n",
"data[len(p_range)*len(n_range) + i*len(p_range) + 0]['visible'] = True \n",
"data[len(p_range)*len(n_range) + i*len(p_range) + 1]['visible'] = True \n",
"data[len(p_range)*len(n_range) + i*len(p_range) + 2]['visible'] = True \n",
"\n",
"sliders = []\n",
"sliders.append( dict(\n",
" active = i,\n",
" currentvalue = {\"prefix\": \"n = \"},\n",
" pad = {\"t\": 50},\n",
" steps = steps\n",
"))\n",
"\n",
"layout = dict(sliders=sliders, \n",
" # hypergeom.pmf(k, N, K, n)\n",
" title='PMF for hypergeom(N='+str(N)+', K, n) as n increases',\n",
" yaxis=dict(\n",
" autorange=False,\n",
" range=[0, 0.4]\n",
" ),\n",
" xaxis=dict(\n",
" #tick0=0,\n",
" #dtick=1\n",
" range=[-1,275]\n",
" )\n",
" )\n",
"\n",
"fig = dict(data=data, layout=layout)\n",
"\n",
"offline.iplot(fig, filename='hypergeom_var')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Above, I fix $N = 1000$ and the ratio $K/N = p$. As $n$ gets smaller, compared to $N$ and $K$, for values of $K/N = p$ away from $0$ and $1$, the distribution of $Y \\sim hypergeom(N,K,n)$ becomes approximately $X \\sim binomial(n,p)$. "
]
},
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