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tanemaki/cumulant_generating_function_with_sympy.ipynb

Last active April 13, 2021 04:36
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cumulant_generating_function_with_sympy.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Computation of k-th cumulant with sympy\n",
"\n",
"\n",
"目的：キュムラント生成関数の挙動についての理解を深めるために、sympyでk次キュムラントを代数的に生成する。\n",
"\n",
"- キュミュラント母関数（cumulant generating function）\n",
"- モーメント母関数（moment generating function）"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import sympy\n",
"sympy.init_printing()"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"s = sympy.symbols('s') # variable \n",
"x = sympy.symbols('x') # random variable\n",
"p = sympy.symbols('p', cls=sympy.Function) # distribution over x"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## モーメント生成関数（moment generating function）"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"# moment generating function\n",
"mgf = sympy.Integral(sympy.exp(s * x) * p(x), x)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"# kth raw moment generated by the moment generating function\n",
"kth_raw_moment_expression = lambda k: sympy.diff(mgf, s, k)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"kth_raw_moment = lambda k: kth_raw_moment_expression(k).subs(s, 0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 1st raw moment"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x p{\\left (x \\right )} e^{s x}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ s⋅x \n",
"⎮ x⋅p(x)⋅ℯ dx\n",
"⌡ "
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment_expression(1)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x p{\\left (x \\right )}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ x⋅p(x) dx\n",
"⌡ "
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment(1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 2nd raw moment"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 2 s⋅x \n",
"⎮ x ⋅p(x)⋅ℯ dx\n",
"⌡ "
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment_expression(2)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x^{2} p{\\left (x \\right )}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 2 \n",
"⎮ x ⋅p(x) dx\n",
"⌡ "
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment(2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 3rd raw moment"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x^{3} p{\\left (x \\right )} e^{s x}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 3 s⋅x \n",
"⎮ x ⋅p(x)⋅ℯ dx\n",
"⌡ "
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment_expression(3)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x^{3} p{\\left (x \\right )}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 3 \n",
"⎮ x ⋅p(x) dx\n",
"⌡ "
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment(3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 4th raw moment"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x^{4} p{\\left (x \\right )} e^{s x}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 4 s⋅x \n",
"⎮ x ⋅p(x)⋅ℯ dx\n",
"⌡ "
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment_expression(4)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAuBAMAAAA/2OGZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAmYkQdqvvRN27IjJmVM1kwppwAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACfElEQVRIDZ1UPWgTYRh+cpeYn8slraNT/BlSHHJ06JqUREQcGtBFlwoOgghmEHFrFseaTBlEbBAERaSZXNtRi9BAN5em4KJTaqyhatT3u+/nvqvNcb1vuO95n5/Ld2/eO4Cv2DlHoBNtlxbqJ/JzszlOVSLE4sMIISDVjhRbj3JEYKkR6dfWnEixH5FS1oEvFrY/8e96zB7pVQBOj3VxPmws7zNWfZV+wyN4o6kR1o6KZTYFv6XpHtyuexiGqWKvJZ3ocmR1ZiRF+3JFK7a82FdJmwOB8n1J0d5qeIVZUbGsd+sLwrDiaM6/WmHPzu5vci3VVZ4FgYqKAWz/kGTks73yPDlxnj2PgnHIC7N28wGd5sNBPzn3pnYNi0QLzmDnzZy+OpEElblvPDaP95UnLjQSq5jDM8KCyw4IP+ymR5KgMj+hC63zKDmfXPQ2V8c2Vj3ObtIP30NsoJnWf7peOPjCARxqdMm5Q5XgrCEQGyFVkARJu8TxpUaaGt0BiwEux2Kn2ig3JEF7aUAXtqxffAeKMPfdQwou2QTKBezqprUZ7t6hD1iVwz1kb7stERxrSbmP53bXM93vu97kJD2Uw3uIdz2cBSQXK1DDK9nHhiSoQ2JIzOvVxY/uDZAc184Aj0gTXK7H8MWXlyVBd/QPCQsaTXbVhksbGKawFfd/SRiVc5/2mFFmoliJ3xKpnZ6eraeSUC+OJGhPqbZL0m59dqF6TdNdKXl7fuhhP5J9xXEfhY223xyyWiqENPptyw1/HbJqhfQdsf05UocrLfFuh3MrV/a/v01JQSBxK0idquULU6UgYaUXpE7RirgxRQmkO9bdQH2K+OJKlDP+A3Y4nCVS1tO+AAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\int x^{4} p{\\left (x \\right )}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 4 \n",
"⎮ x ⋅p(x) dx\n",
"⌡ "
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment(4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 5th raw moment"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x^{5} p{\\left (x \\right )} e^{s x}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 5 s⋅x \n",
"⎮ x ⋅p(x)⋅ℯ dx\n",
"⌡ "
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment_expression(5)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\int x^{5} p{\\left (x \\right )}\\, dx$$"
],
"text/plain": [
"⌠ \n",
"⎮ 5 \n",
"⎮ x ⋅p(x) dx\n",
"⌡ "
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_raw_moment(5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## キュムラント生成関数（cumulant generating function）\n",
"\n",
"An algebraic expression of k-th cumulant upto k=5 is available under the \"cumulant\" entry in the Wolfram MathWorld (http://mathworld.wolfram.com/Cumulant.html)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [],
"source": [
"# cumulant generating function\n",
"cgf = sympy.log(mgf)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"kth_cumulant_expression = lambda k: sympy.diff(cgf, s, k)\n",
"\n",
"kth_cumulant = lambda k: kth_cumulant_expression(k).subs(s, 0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 1st cumulant"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\frac{\\int x p{\\left (x \\right )} e^{s x}\\, dx}{\\int p{\\left (x \\right )} e^{s x}\\, dx}$$"
],
"text/plain": [
"⌠ \n",
"⎮ s⋅x \n",
"⎮ x⋅p(x)⋅ℯ dx\n",
"⌡ \n",
"────────────────\n",
" ⌠ \n",
" ⎮ s⋅x \n",
" ⎮ p(x)⋅ℯ dx \n",
" ⌡ "
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant_expression(k=1)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\frac{\\int x p{\\left (x \\right )}\\, dx}{\\int p{\\left (x \\right )}\\, dx}$$"
],
"text/plain": [
"⌠ \n",
"⎮ x⋅p(x) dx\n",
"⌡ \n",
"───────────\n",
" ⌠ \n",
" ⎮ p(x) dx \n",
" ⌡ "
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant(k=1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 2nd cumulant"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx - \\frac{\\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}}{\\int p{\\left (x \\right )} e^{s x}\\, dx}\\right)$$"
],
"text/plain": [
" 2\n",
" ⎛⌠ ⎞ \n",
" ⎜⎮ s⋅x ⎟ \n",
"⌠ ⎜⎮ x⋅p(x)⋅ℯ dx⎟ \n",
"⎮ 2 s⋅x ⎝⌡ ⎠ \n",
"⎮ x ⋅p(x)⋅ℯ dx - ───────────────────\n",
"⌡ ⌠ \n",
" ⎮ s⋅x \n",
" ⎮ p(x)⋅ℯ dx \n",
" ⌡ \n",
"───────────────────────────────────────\n",
" ⌠ \n",
" ⎮ s⋅x \n",
" ⎮ p(x)⋅ℯ dx \n",
" ⌡ "
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant_expression(k=2)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )}\\, dx} \\left(- \\frac{\\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{2}}{\\int p{\\left (x \\right )}\\, dx} + \\int x^{2} p{\\left (x \\right )}\\, dx\\right)$$"
],
"text/plain": [
" 2 \n",
" ⎛⌠ ⎞ \n",
" ⎜⎮ x⋅p(x) dx⎟ ⌠ \n",
" ⎝⌡ ⎠ ⎮ 2 \n",
"- ────────────── + ⎮ x ⋅p(x) dx\n",
" ⌠ ⌡ \n",
" ⎮ p(x) dx \n",
" ⌡ \n",
"───────────────────────────────\n",
" ⌠ \n",
" ⎮ p(x) dx \n",
" ⌡ "
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant(k=2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 3rd cumulant"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x^{3} p{\\left (x \\right )} e^{s x}\\, dx - \\frac{3}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right) \\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx + \\frac{2 \\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right)^{3}}{\\left(\\int p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}}\\right)$$"
],
"text/plain": [
" \n",
" ⎛⌠ ⎞ ⌠ ⎛⌠ \n",
" ⎜⎮ s⋅x ⎟ ⎮ 2 s⋅x ⎜⎮ s⋅x \n",
"⌠ 3⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟⋅⎮ x ⋅p(x)⋅ℯ dx 2⋅⎜⎮ x⋅p(x)⋅ℯ \n",
"⎮ 3 s⋅x ⎝⌡ ⎠ ⌡ ⎝⌡ \n",
"⎮ x ⋅p(x)⋅ℯ dx - ────────────────────────────────────── + ─────────────────\n",
"⌡ ⌠ \n",
" ⎮ s⋅x ⎛⌠ \n",
" ⎮ p(x)⋅ℯ dx ⎜⎮ s⋅x \n",
" ⌡ ⎜⎮ p(x)⋅ℯ dx\n",
" ⎝⌡ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" ⌠ \n",
" ⎮ s⋅x \n",
" ⎮ p(x)⋅ℯ dx \n",
" ⌡ \n",
"\n",
" 3\n",
" ⎞ \n",
" ⎟ \n",
"dx⎟ \n",
" ⎠ \n",
"────\n",
" 2 \n",
"⎞ \n",
"⎟ \n",
"⎟ \n",
"⎠ \n",
"────\n",
" \n",
" \n",
" \n",
" "
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant_expression(k=3)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )}\\, dx} \\left(\\frac{2 \\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{3}}{\\left(\\int p{\\left (x \\right )}\\, dx\\right)^{2}} - \\frac{3}{\\int p{\\left (x \\right )}\\, dx} \\left(\\int x p{\\left (x \\right )}\\, dx\\right) \\int x^{2} p{\\left (x \\right )}\\, dx + \\int x^{3} p{\\left (x \\right )}\\, dx\\right)$$"
],
"text/plain": [
" 3 ⌠ \n",
" ⎛⌠ ⎞ ⎛⌠ ⎞ ⎮ 2 \n",
"2⋅⎜⎮ x⋅p(x) dx⎟ 3⋅⎜⎮ x⋅p(x) dx⎟⋅⎮ x ⋅p(x) dx ⌠ \n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ ⌡ ⎮ 3 \n",
"──────────────── - ──────────────────────────── + ⎮ x ⋅p(x) dx\n",
" 2 ⌠ ⌡ \n",
" ⎛⌠ ⎞ ⎮ p(x) dx \n",
" ⎜⎮ p(x) dx⎟ ⌡ \n",
" ⎝⌡ ⎠ \n",
"──────────────────────────────────────────────────────────────\n",
" ⌠ \n",
" ⎮ p(x) dx \n",
" ⌡ "
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant(k=3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 4th cumulant"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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vX25o2mJTF9mZk6KPWwg6JZwFlTfD2/u6CrvxULcQku4kXt/C6rNqJUxgMj4UwXgWqA8FGzQjX00iUTu9BGvLpHeIKDjMYNA0ZWuGWVDmlp1PwMUvEPzmJLQMJkpo1eg8lBudkJOIwBsTGV72givJIVSZQVPZ6UrlykGBRbRbCDolbELh3AN/SDFYbiFwyff58TFeIAkiVkfo3qtbWH9FigKT8aEIxsPOCuNDwR5tm6jK0iQST0LvwkjhuD536NC3Dh26lYthxSPlb3NNaGbDLJpbUhmbC4Bvcb3qMINmAs2gPV+yPokm8CIBG7zYg1gh3rfZrmazeP0yqtjSbiGzbZ62SZ6sZRn2tsy4hcgk919j2nlhcq6RY2KLyehDEYTH8qGgg8bWJDCepN6Jtif+ZHCnUTwaZo25BZXxufAfNBNpls5DtW2zPCNpyOQkmsSw46Axq2dVvRfCbKUybiHSKeEbMNuzwSxStxA7K3mM3S7UcWRVCQITW0xWPhTKSWIMj+1DwQbNyF+TSD1JvUNEwWFqg0Z/wQDNbJg15hamjc/F9wn+24cYsQ0mUo1q9Hw5IScRIuLhxpX6jBLJBi8tFVlGNtmOBFx493ELQaeEu2CqB9Am2hepWwhJdxLN4+rRZalIYtI+FIF4bB8KuhBgaxJa0eklUBv/b2/j9OLUpViVUxs02p8FzWyYRXOrzsfngi4EmNWzCTQbz5cTchIRxosdz45YhZf8bT0pHS4im+z8Us/c6KDA66JbCDol7GP/Wgxv7xK1llsISRdiUlArfdR04ZyQFCb0oYBgPJYPBevUwF+TSHXsHWrGMGlvsb4JUxs02p8FzWyYRXMrKuNzcb7BCy8EGEwUQdV4YhV7sU4i0kxSsdjz7A5UeMnf1gufG67f7AIObg33V7BnRbUhAH08WB3tFoJOCewRf//7O1Sb5RZCM7icFNTaUGnC1/gKE/pQQDAey4eCvadZ9tfEU117pzTrIGlvtQItpDZotD8Lmlkzq80tqUzABU5MOOoXugq7x2A8VavGE4v5FsU5iZRmt8Bzp5F46d/WVwbYwOUoWCHdBQzmD4C2mlYpHmHPEOxoICNAX8bznJAjMah5HDQz9oaAw6LBY7rZcDzskUseHk080bF3SrEOEvdWa9BCaoOGbfNoCa1oZs0s2OaOz4X1j3sXIvQINNN9B1gt83ChbzUh8daXTaLPLmDb34fsAob6q1hxZQklHcoNKWWtu9rWWVGExKD0tsJpulcD1I6W8Q2bwWCCNPEajr3zNpq4t15FAL80npQ0RfmzoJk1H2CbOz4X1oZNvfDqMRhH7VWNGzaT9ihJvemP2LUk3qm2SfU5qSx/H7oLGIp6BO3tGBVKYs/B/HhKRc1WfEyYHCYGpSdVhR/TFrw+FBHwzMseMGd+S5PQ6tg7iozLiXvrVWTFa0MrGj3SFkWr6oKHZtbMgm3u+Fzk9InD3nD/BHFFoFm7BmCdSOE/RirlU6gt0uzpmcI72zHlzfQFdwGD5e8jb8WqPPrBsnlp36iQ0phbiB493pL+8cSgGqtKYUPfB3mC14fCjGb/9lmqXi2yNfHyrr3ztpm4t15FVvxxHqtc/ZCVGCXyVVEIL6BoZs2sbe4EXFTVa2feTO01RBSB5noXC8cIawNeODkP9h/LKbzrQ42ATvTVvl2w/H3k/VKVz6nNXZYjkdYlhUpHhlEWFmjVxKCK20oNLvpQrRBnoUO/zPPXxPUm7Z2NyfoLx3gm8Cqy4h/isanWeSYxogdJtcOqaH8WNLNmlvqNGd0xuCigjVht/NsOtvBCUo1aLqFq4+5s50+OlVo8n/IQVR/nYaFP/2FF493qc6VBu4Cpv4+9C9j8TZW5WnBNqRyJQem/b9wcOALR66LOmkKBJO7tJM3FNs+9GT6gC+llnkJXpxHB+J5/m6Vqfxb8NzLNrLO5zWI++7MYPffNiuarRRcpDw9gpyPw0OhOn43FeajwrgkGg3YBE38fzy5gyOM+gGl9y6HqneTEoGo4X17pOQEA8x/frprCcSTu7STV6+IK+yQpopd5zLMB/Y4naN/z/ayS9mdBM2tmnc293jaginoWnRHNyiGS8nAbNh+Bh9rGX1pXGIX3CCc3cBcw8fdZsHcBs0cZ1XjhdUSRWpgc1DkKw96hGxjzj0WumsJxJO/tBN0Pi7yfIyXu1PKGkuzveG6JYcay1nVBLqCZkVlnc6+Q07C0jW1lRHN1WTRAeDBLuno1LhIPQo/CK+buCXYBA1zflx1u6I4jAc6hAyjVNpo6KZT6CGu6akI9wWHy3gbrVF8n1LMfVtIs8yyoevZ3PA+jtkYHJR6ima9Xic7m/grRPjvCSEY0T82JBggP5uUhuk9DJB6EHoX3EyISfxcwe/vVEVVhdlWGKf4mB7U1lDBw7pgU1GIba7pqQj3BYfLeBuuUXyekf49eGenSym0Y7O94au9se5UVzYzMuprb+gP0zWUElRHNmy3eAOXhQWyRvQhSJ0skHkQ1hVc+8FZj7wJmd/El2fxmT4Yp/iYH1ZBWqLk+Z60IslmXnDWF05K8t8G6y02eNyMfGfQyj3EbFp8kJN/xpL7n06Iur88PNLNiFlzNPU3/hWVlTjaSGc13CP2SB+OZnYQHoUjhlRuB2GKy/Cf1wH273l3AbBlvJDu81pdhir/JQalFnrJ+wEyICmfw4KwpHEDy3gbrZt9KY4f6EJNe5lH+6GX1SULyHU/qe269w9ZmxuUzV3NPLXFk6hAzUy5nRfN7REOSBz/3afFpxkg8CEUSr7pZxt8FzHZgvSL0wDtkkOKvAyh4i8BRaLrBMZ8PdNUUjsOlt4Ha6wOeVRVXdbPMo/zRc+qThOY7npZ3tvGKEjqUmRWzzua2Ph+oF/azovlp3gXFg4/7tPw0YyQehCKJV73wjL8LmE0U1RTIdSYk0Fg/DqDUPsPZnqUwdsS8xnXVFN60S28DtZeWeJZ8j2iWebTbcIxPEqKZcQenq7n1oxPDN/MyR8mPrGgWzt+KB+2ZbdynY/DAUSq86qoiV07M+gp5auSF/Q/1AOs6ExpX7gJKnu7yfeG45qgpp+MjDbhqCm/RpbeB2uUTSGkkCpBlHnGW1mJ9klCaGfdPqsekwIbDMnL0ZWFBP99kRfN3BZ7SSARsUd3t04wKr5q/yuF/p1DNf6ptLQYLfHZbHRUGwSUS5riAqrAxfBh+M2HLqlr+AiU4a4qAw6W3gerlnUa9fDfLPHKAzsT6JCE+xHBm3c29MCSYcRdodjTLO43iYdx9OuanGRVe6ZiWYBcw6zq/pdabj3UJC+mILqDgLIAnam91A1JfVvWdNUXA4dTbIP1y0Cgj62WewA/msb3Kxjvb80lCPXNizLqb+0mKGMdjdjTLQaN4QM9s4z4d89OMCm91lXciwS5gVos/ZlY2vkxZSEd2AQXFJbj/9/puQD6N1Z01oaIJoVNvg/TK+79624TLPMZtmG1Ky19D6lq+5+rWh9lyNYHFGLPO5i4PUC0PD6lIdjTLKYfiYdx9Og4PiNf7t/W4D7Yc7e5xnupzuoEjKNjjDGdGXEmc1URS4NrbgEbku5b1OZGNyzzEbTjOJwm1md2Zha9TvK7PR1RXgHyDSJc8+LlPx+FBvfZy/Nv6tVYAVKdkR1BQHDo1zypflEm//FG59tZfq/rE2mZfZHuXeR4MqKSS7R0B7KNWSIc7s/nLaNP4vpSmpSzLDXWSB/Zox45GVzcRkweQeB3/tr7R0+2nKDiCShHJiVCVVW/FUobag+ld5tmY3LFS287PxsysjXkcjXZ7acamhlyb5EEurJaXtf6YPKSD1/PEqMHsCiefgX0cwprA4V3mMVvi/WEues7lzMx8rn/7aaYWe1yb5MHrmR2XB0gH72c4ot3jVGRAfDbvcwKZd5mn3p0M+JPe7IzMXFjyNpR+vNbkOgUPY57ZcXlICW/B3OrS7++uRhcG2GfzToMvWRpwmeceK3U88rw3KSMzZzQWbfS/Cl4eKh1VIi4PaeH9oI1wN3bKMMDmHh/Pj5LAKffGamVi5orjy7QxlL4Jl0BaPKSG95d9ge4mngIMvAm+sD9kHuaP8oHx5EzMXEmEbhzc5JTcclo8nBi8k3uzm5stA4k/m3dFtrhOtPafTdrgG4yHpDTs1ttlwJmB/wMqVU0iGZAcegAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x^{4} p{\\left (x \\right )} e^{s x}\\, dx - \\frac{4}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right) \\int x^{3} p{\\left (x \\right )} e^{s x}\\, dx - \\frac{3 \\left(\\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}}{\\int p{\\left (x \\right )} e^{s x}\\, dx} + \\frac{12}{\\left(\\int p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}} \\left(\\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}\\right) \\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx - \\frac{6 \\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right)^{4}}{\\left(\\int p{\\left (x \\right )} e^{s x}\\, dx\\right)^{3}}\\right)$$"
],
"text/plain": [
" \n",
" ⎛⌠ ⎞ ⌠ ⎛⌠ \n",
" ⎜⎮ s⋅x ⎟ ⎮ 3 s⋅x ⎜⎮ 2 s⋅x\n",
"⌠ 4⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟⋅⎮ x ⋅p(x)⋅ℯ dx 3⋅⎜⎮ x ⋅p(x)⋅ℯ \n",
"⎮ 4 s⋅x ⎝⌡ ⎠ ⌡ ⎝⌡ \n",
"⎮ x ⋅p(x)⋅ℯ dx - ────────────────────────────────────── - ─────────────────\n",
"⌡ ⌠ ⌠ \n",
" ⎮ s⋅x ⎮ s⋅x \n",
" ⎮ p(x)⋅ℯ dx ⎮ p(x)⋅ℯ d\n",
" ⌡ ⌡ \n",
" \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" ⌠ \n",
" ⎮ s⋅\n",
" ⎮ p(x)⋅ℯ \n",
" ⌡ \n",
"\n",
" 2 2 4\n",
" ⎞ ⎛⌠ ⎞ ⌠ ⎛⌠ ⎞ \n",
" ⎟ ⎜⎮ s⋅x ⎟ ⎮ 2 s⋅x ⎜⎮ s⋅x ⎟ \n",
" dx⎟ 12⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟ ⋅⎮ x ⋅p(x)⋅ℯ dx 6⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟ \n",
" ⎠ ⎝⌡ ⎠ ⌡ ⎝⌡ ⎠ \n",
"───── + ──────────────────────────────────────── - ─────────────────────\n",
" 2 3 \n",
" ⎛⌠ ⎞ ⎛⌠ ⎞ \n",
"x ⎜⎮ s⋅x ⎟ ⎜⎮ s⋅x ⎟ \n",
" ⎜⎮ p(x)⋅ℯ dx⎟ ⎜⎮ p(x)⋅ℯ dx⎟ \n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ \n",
"────────────────────────────────────────────────────────────────────────\n",
" \n",
"x \n",
" dx \n",
" "
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant_expression(k=4)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )}\\, dx} \\left(- \\frac{6 \\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{4}}{\\left(\\int p{\\left (x \\right )}\\, dx\\right)^{3}} + \\frac{12}{\\left(\\int p{\\left (x \\right )}\\, dx\\right)^{2}} \\left(\\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{2}\\right) \\int x^{2} p{\\left (x \\right )}\\, dx - \\frac{4}{\\int p{\\left (x \\right )}\\, dx} \\left(\\int x p{\\left (x \\right )}\\, dx\\right) \\int x^{3} p{\\left (x \\right )}\\, dx - \\frac{3 \\left(\\int x^{2} p{\\left (x \\right )}\\, dx\\right)^{2}}{\\int p{\\left (x \\right )}\\, dx} + \\int x^{4} p{\\left (x \\right )}\\, dx\\right)$$"
],
"text/plain": [
" \n",
" 4 2 ⌠ ⌠ \n",
" ⎛⌠ ⎞ ⎛⌠ ⎞ ⎮ 2 ⎛⌠ ⎞ ⎮ 3 \n",
" 6⋅⎜⎮ x⋅p(x) dx⎟ 12⋅⎜⎮ x⋅p(x) dx⎟ ⋅⎮ x ⋅p(x) dx 4⋅⎜⎮ x⋅p(x) dx⎟⋅⎮ x ⋅p(x\n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ ⌡ ⎝⌡ ⎠ ⌡ \n",
"- ──────────────── + ────────────────────────────── - ────────────────────────\n",
" 3 2 ⌠ \n",
" ⎛⌠ ⎞ ⎛⌠ ⎞ ⎮ p(x) dx \n",
" ⎜⎮ p(x) dx⎟ ⎜⎮ p(x) dx⎟ ⌡ \n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" ⌠ \n",
" ⎮ p(x) dx \n",
" ⌡ \n",
"\n",
" 2 \n",
" ⎛⌠ ⎞ \n",
" ⎜⎮ 2 ⎟ \n",
") dx 3⋅⎜⎮ x ⋅p(x) dx⎟ ⌠ \n",
" ⎝⌡ ⎠ ⎮ 4 \n",
"──── - ───────────────── + ⎮ x ⋅p(x) dx\n",
" ⌠ ⌡ \n",
" ⎮ p(x) dx \n",
" ⌡ \n",
" \n",
"───────────────────────────────────────\n",
" \n",
" \n",
" "
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant(k=4)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 5th cumulant"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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sJ6F2dvQPWrq6dpfpssQ7OMDdYoxDhWskuhVIuxPLlIyWw2A28gj/7MWLFwf437DXjUijo7iNWcJdqpSyQEUYzfiPNH43GFGcOoa3d5lCAWjOkuXOX3vDfmZVK5B4wucGBPmApudb8qscU0Azo3LWXO691+zk1Q83bhKkoe1VFUVzO81u9Q7j7Q1N2LNGu8sx+aE5x5B7mQg56Jjw2TXhr8uzDD1r4pJ8s10r1CJtheb2Wo8rdevGReTf7SmUuxjttex58FmdFvxHdWU2mOXoUHdyTyNsrWovexp77Evm87kf5niAc8brIN1hxu1jpA6nyujQEvWWvu4WY8kSvQrUP+8aDbINxgxiutzGrGr1qMrSMioiWTPm/GfD1aljMPMkGdDxW7ChJHxhwm12zUNCxlkadXiWFH+1pqOAZkblrNFdnr0wtMGrKlGkbqfJrT7JDpLpwrJ0xMmzRqkOpcE8z/XiessKxEVHcXyRg/Q+rhzguSsdK8AibYXm9s5+RyedcpYAr6IXD0XsezjJlez5rYs7ADqn3mCWo0NdfMGMj5Nd9ml3tsfeFD7WOw5b+5wWbKRa14zbx0ed+eXWEnVTcLcYg0SJbgXS7sRCvkaDbINpc0X0cBuzqtWjKgvLqohEzZhTbi9FdO06xhCajERPFnT0Fkx0+Bwo3GKndnoRf1RUh2fJ2zm8iGa9f4zOHNlzhrqsRdvpjqo03U5TWv2wHSjjwqGm61qjXIfSYJ7nM+L6kyZw4uFb5YX+gZzTA5NJZPt+xb6wSNvBef2Ter4A2s4S4KWw3SEQo77ZpZCJ3lQX/xzUUoXYzw4qKMzZ7er8D+9rL3tM00zmYy/bpiBNXHvto5uEd4zUmR2QjUR+9HK3GEuW6FSg2p1YiDRo2QaDiS7ZJeKZ25hVGQ5VKiqrIpI1Y97LVTtFr91acHabV6+yoJ2nX/iBiNIJKky4za5F91bKEeiuJdWm5BheRLMFR4Rjn1UV2eXYVaWKM+00udVb44yNBNy+EMa1RrkOldYFphbhGfCFYdM0NlXYHcNxTsjX7SuLtB2c128N+ytmCMfc94gXXOTR1j8jtWUVsjO9s90cNOkKXz+PfCwtMqD4W5Z/XNAByXyaA51A/DVEHR4SfeYN446POmsHZCPxhKJvbTGGIckSnQrk3YkxtUHLNlhao1U0Sjncxky1gkOVQLMqIlkz5n3YtE+ntVyAt6jGkwXttuD443aYcJudumO1Xpt3Lak2JUerFNFs3weXquGsTOqRyKLtdUcJYNppcqu3xhmHcVs/+bvtqmSHSusC+BTw3togS2p6fL1vx1mk7eC8fu5TmH5p2c7UvLevLq1V0vdRUP2D74Qr3nn2HpXgFuVGc7Z2GGrCmrxUYY0BRwIk8qHlAkq0sMqJPaQPfXORYo6POrMDsiVxdp9oWluMYUCyRKcCeXdinDYcEIQ4ZxosrdEaiOI+bmOmWr22JiGzKiJZM2Y147Zbx5+9+bWKaxY0/oHCOuKP22HCHXa0BlX/rubrVDqoTckxtpBmb2VLg8fyqHHbou1WlSzHaqfJrd6uJZuxNQQ61ijZoVK7wCXw2VO98ia5y8lqkXbCc15wn8Lk5/Yoj9pMFi/MBrNyv1kQt+C1sy53e4NZyh9+Nq8LNR+3OX35i7/zts7MGgal87nz7O4iR/Mr8wLDRRIhx0ed4QtaorelryDMR5pErkBlHEyu0biEhAp00VIbLRdewuU2ZqpVtzVvH+Pi1Yp0zuwwpzh1HH/cDhHu9S/qD9aLhVzpQb1W9TK2Y3RXjdsWbbeqdDtNb/WJdiCis+ZXjK2R0AXSoSFnF5hQ91AlzeO+RWeRLoO3cqhznemTV20mixd6g1nabxbgA/iaea+h0lFqe4NZCgk/n+sxxqz6UwVx6v/0YnNvbhN/POT+t0l85i/FDWs4GlfyN5KRROjxUWf4gpbobenLMtBNlagqMAmNw4Yr0EOrYtzmNmaqFRyqOSrCY2mqFe1xXncnt7UIk+WAdrZIFnnij9sBwn12PyoYQtPMDETptYQqoas41dS4bdF2qypHq/fGGZsxv6+E1J12JaXkgPYaVxVdYNiqFunhyOyQlWWd5uIiedVusXihN5il/WYBngON58JLd3QWHEHtDWat8CDvjR3OvqBeIpCcGovvhvphG2mm8zmxif/g4mgctweJSDLw+KgzfI3E1C19IVWiqsAkNA4brkAPrYpGy23MVCs4VHNUhMfSVCvWpBm3ndZSuo5/gJtMNDdAuN+/XiJJWRtFROm1hBpNrw9UU7OdFm2nqnK0+mQ7UEHWEOi0KxGbA9rvUFV0Ad8i+L2aJwD/A1HmkFjcp1DhxR6hq91i8UJvMEv7zeIr3TD1gpt+2SbhbDALV/6TOP6zSFCGEeaR2Cd1CdvqxkJyquHrb81HJvfx10LtrTjMB7vI13Q0rs4sM5SDJAOPjzqWI4SxRFULraEtfSFNIldgEhqHZRisefr0L/3o6dOrwj4lK9DNdiiQuI2ZanWp5qiINM0C3ozbEer4k6dPf+X06XcLXFdJyStpgADhbv8CtY3clIQVJNmSENBrGVXgWT14oaRkN5sA5XHb0AanqnK0+mQ7CHA5BJJHW4Obe4kOZXWBOAZwzSGuPNKKezGHn2Fxjyv1MIP5v8UYB3qD2YWuCNsVe8A5B/6lYAUHUpPFiS15Ydr5km6gghO+Cja3LMZtnLjuivMwnxdDTbzOSNH2uO0iiczW9rlpaEemjvnaEmUt4LhtL8EI2mkSTQUmoaVJ9NCquNngNmaq1aOa3cw8lu1lYQh1mHE7Th3Hv98OEO63wK9I0WYANJY0XTDbnMmobNDoLs+TGNrgVpXVsVNa/SjG5n7bWCOpC6RA+x2qii4wbFJDejguRwivGeHcg/6/sPmejtlglv6X/zew0HdBV+wNZt2o8ld4d6WOcxvKIznhq2Ara2KeBEfsHXEe4oNfiZ041NE4bg9EMnG4SDLo+KhTcpCVkZi2pW+qRFOBZBwHLa/BIjZavRc0tzFTrR7VzIpI1Szq8cy+OIsjTh1HG7e1AcoLdzdwRon/JIWaCQdjyZBeS6gSOuZJbwGvbgkNba+qslu9P87YjGfFnZo8jDUSukDeDhWxCzAvfCYwXvIZ0n5MrmvuU4j8qMqgdosVV7zBrNpvFtclJ/sAXRGljhV7g1kODHXNOs75NcIiTvgq2G0dsS6Zyqf2DZjbeDZH2+uSLpJEPTbqDF/xhh9JTN3SN1UiV2ASGocNVaCPFrHR6r2guY2ZavWoZlaEz9JZlzwjf8JFjcap42jjtjZAeeHg9y9/XZItCUG91l7lk10jyslsAa/GbWtd0q2q7FafbAeiaV7NYGtwc8f4bGi/cUXsAsaMSx3jlz5F+s6b/QgvXcol9ykc4tQSIO8WKzLwBrNqv1m4HmY24KU9C8vZYNYKl96ypNZ1T7x8UQIpTgt7rRfJj92n87kVPrza52gUtcecXCQZelzU4eff5Xa+ghRL5FqYHtrSF9IkcgUmoOkShivQQ4vYaPVe0NzGTLV6VLMrwmNpqhVN9nBKaxHmLNOCo43b2gABwv3+9WKpaka/+caW5PZSTjOhSuiIp+l+/ZDg1LhtaIPbHbNbfbIdCL09ZI2ELsAGms7sUBG7gDZm7VxH+x3SzxM3wurgva/qPQ5Jd83fLfmPV7xbLObRG8zSfrMAuOJ46uc3bTRng1k7QvjLktraV0j8ryrF6eAvTvVgfsD73ybxueLm3716n+niMLiWjCRCY6prbFI5eUw+pE5t5ysRWCLXwvCWvpAmkSvQqAdG0yUMV6CHFrHR6r2guY1p4fpvbd4+xsw8k6WuVmGzh7mZe61FxJWp42jjtjZAgHC/f9FDsdlQhCud20s5zfyoLUwW8+CtzdT/3A1tr6q4nbKK4VafbAfiav7CwtbghoTx2dB+h4rYBbQtZ1Y62m+TntmY2eGIxh77Xssex7U3VoO2/nP7wbKTSlzgtBMebe4Xzj/vREzGUZrU0r5Cnnf/Lnm/DL6gi20MtDfFg9NidHhIIjCiOtD/hcph8jR1kqgv8W6iP+KsJSZUoI+WbbCojVY9HHIb08LBpVq8IrRmYZfL2TiR6jjauI1/QabeGk84TvSLY0LP6LqWlJHFzalQZe6oJ96e/WGyg6HtVZXfTjNbvcN48pBJJ1ijMDRat8t40dz/MTRuE+m5NVPEbI/9evs3d8d0a2M1mNPPGOurnE27ODeER11jN7s6Ko+nNKldLmfGfZfiFlnq7brsbD5pSAIiorqEzdfSTT6Kk/pPs5HIO+xozUMeDZdQgcUNBjXdeoZKKhzwVcrBbUwzBZdq8YowSFiCfg/Kay2i8OLQAD9OrGOclQEM3VDhaj8ls62SCyg5F9ds4GKINhh6C/jzOzLQlONVld9OM1u9QUJgnocDr13JMgtD47uDEbuAMkZ/aNwm0pNdlQAd0+n1uOZ858DeWA2m9SB+ctNAKB8uBojji+rSbAPKAaPd0qT0Y/DUt+0ScAFBHHParPp3yU7l+JdIAX4dzkGSaSKqS/hpSzd5ijrJyZeod7R0VDkXWuJwBfpoRSuwte+UlP+iK5M21ygHtzEtHFyqxStCa8YCat+jUqqo49IGmHEMEE8478z0KGt2LSlDi5uTURnUuP/beAv5SD/wPMmNqhtq2m539Ntpdqt3GK8MmNuwNYpDM5bnlrWEfIuj3hsat4n0wqYp5oL28vZv4HznoDnQCezPcz68YwVL79AGs3oA91MmX5cm1d5QgGbFQQQ857GucMw8UDYfvZLvIgmYmOraPYEojhwmT1Ens/sSzW+UjE46aYnDFeijmV/gJKDhsM+JoMZ19wzHZIT8mYznGwj+TpkWbi0lYsISFaE1Y/bW40wmfh1LA0CJ5fX5VUGKDRBPOL+R8Q+sebjSS5iTURlUu6094S2tX28Bv94VOACatltVfjvNbvUO4909CY6nYWsUh2Ys1yVLlO0LL4OhcZtIb+/rYuw5zPepUOc7B/QcpWImaMIMr/TjpkZiT2OTfHkW3TiPcEuTmj5UMAvssWHFl5TlkYPPSc6YjCRiI6hL2HwN0k2eoU4yZol3MP90V0tMr0BGy2Ewp5xfFFeTnReaQEbK4NXcxCx6L2huY1p4clsrUBFaMxbDXx22n5cNYfIVgHaySgOUWl7/TcTRBognnP+5ciPTTK/0ApoZlUG1O9sRXvv1gryoQr/ZAn6pL3AANO3k7pizdSGQw3hpTYLjKd0a+aEZy3XJEnZfYEtAjr5wybU3vtHFAyJ9sCOC9bZa3vZv9ncOvL2vWjyDbG5bvALKX5YmVefFUvNLWpKFfsEpGGmIAKlL21RwhMnjqZOcWGL8CpzuigLeDu+Q5YhTY095p3rK4zjmS1J/h+F6L2huY1p4MFXWLEqf+TZziF7HZIBSy+ttHPG0AeIJ5xcgtxBeHMGWlCiMKi/s03XiwtafewFe6DdbwO+uEirThtCqchivbxJ6JGsoMNeRlnD6wl2cIE9fgG1VYZwJiPRWTwTova+87d/a5jsH/t5XNf6X5Iy+8dbQoZ7SpFo86b7eD+PQ+BbnD0ViHOOSurQN50aYPJo6yUVLjF+B1NY+bySbL56ZGUjni9sLiyrxKXT1XtDcxrTwUKpasyht2qys98V1xIMMUGp5Xew2rA0QTThsd0nerjJ0qCUJjVF909HHRWz9uRfghX6zBfzKhksbQrujw/jGHUU8jjV8K4hr9ZkVuy+8h9Pl6Qv186/k5Mol0uc6AjxtvzzrOwcn3P3ycEpI4Uw94QGHX5Yn9XxV+Mn9MBZmw99QpGEeUl3qlomjTB5LneSkJcavwHtlAT9iSb9d+88qn/ud+QPRDMWxTY46cxtj4aFUtWaBP3vIZUWvYzJAqeX1hmYl2MUSDus9Eju5Sm6oJQmFUenKnJtrwm/rr8sQEdrsijMeyQvwrv62ssbkqsiBR2hVOYzPECbOnsQfwxQ0WQKsvmAsAYX6AnMl0nLCz2yrxfthZWz/JjD4x4pNy7gR3ABSqnTu8GXJzA04ZygS4xhXqsuzZeKN7haFZpYvCictMX4FPleK1dMQeHVBy+cvSbnfmaeXZTFRe1OnRA+3sRtVYChVrVngLcf3LhkAAAnMSURBVAwUqr4F4etglwywsGmAjAGyFqD/n8lkajxUOPyxQuUZ82BAiceoNmXhn1wUZ1t/uydCxFFIPz8TMe3gqnIY6xeK4liD9LlnsgRYfcFYAgr1BcYl0r8kL61ttRa6IiRzvzz8a8CmSIi1s0FuxHN5Ugf7RIMnccqSWulyzlAkxjEuqcuxqaC3lx3+hSuSOslFS4xegS25ytCw7mEaAy1/VmlwvzNPL8tiIn4BjNJzG2PhoVS1ZgG/u0aFAMSuYzJAyeX1G5iVcGMJr/G837yqlVBLSpIa1aYs/Lsdcd5WVS38d4sTHe9TbsoCvKN/Qg14TDu0qhzGLf12YRRrsD7HJUvYfcGyRKG+oGAVaVqoaRbeLw9rZZWQdvsKMZ5TnlSb+mIrdM59XTY8VBSMNGwVtTZWeC87vBmNpE5y0hKjV2B9WRQwT9PHesnb+9yT9Z15+0tSMzKvyC8ObmNKOIRS1ZoF+PqiOOMRvY7JAHCwI9C1AXKu+b9FZOIjlvAZXrCHhwg61JISxaAyX+XeJl3SX3QB3tHf4LcfFO3QqnIYz/BqWHC78uRbl2QJ6gvGEmX6ggJVpOknB9/wK7hfHj4LDQhpS7ZOi2q4tzwptQBvfUe0HBueV4RgpOHy1a984b3s8A+o1PvicNISo1fg9IYQXf+OlK6XvL3vp1nfmbe/JOX8t0u3MX6vIpSq1iyo0XyVYKoXKCXj8BMZAGgBWhsg55q/nuoXPGIJn1wVaOLY6pATo9dOrkqs4dMbZBDpL7oA7+iHlyhwRTu0qiZXFZ5wzDuBoe3KAvW8ZAnqC9oSUKYvKGAirZ4beFutAtu/6U/VvcpjGn4ZQAp+WBY/tRzGYoLGHAQJRRrmwY9qhfeyQ6g46iQnIzF6Bc7tiRKaNFuil7y976eZ78w7X5Iym0FLDK4HEg6BVI1mAa6eeyqoYzIAnBNDZOE1/11nRI0jHJZ6QrE4JtekE2hJiWGh0rU+Pyh9Un/hBXhX/zlFXNEO7Y7GDshwco8JR7EGgzkuWYL6gv6sHpTpCwqWSDfoUYH3viqw/Rs01FyEchy2YRcBpNSGPAv9MAbmH0ahSMM8lLocG875O+7xdkNROBmJ0StwdlWoVjcG2d9Pc74kZc8E4qDH3JY60pB8KS+Kn4xmzDvPU75Dn/QoDuzlIAPQDX3hNX97VphrHAKFq91rBM0JemYLBZSKb/F068t3S1+5BXhX/3afQBXt0KbvMD7oEjYEm5dxhl2yBPUF/Vk9SP2W4Ii+oLCJtLq9uV8GmjVvax1pmAqHqNXD2A+ZON9IkKVIUdekmTUmWtx9Jg0TmDEUabhspQ6XFPBoq9sJHKLktQgbccRRJwvQEtU89IhSi0bRpOzsQOazlrzlQNka+n4a3kGs7GNaGa1+1WRWPKk1XLXNUyhVrVnAT+kp3+h1rGalVQcp+I287UUWL1yq8VDhE88zmHLoCgWUcDaqwRe+h+Slqjyc+HgALhMhC11xznjnwdM/EFnwoBE3sKpcxie5l0exBtH0z2SJ2YEMN5agxl6sLxA0kVbTiWSU2ykGz82u9qZ7xKRQczC1l56iZEwIqQaO+ffDT5UsWWWryVaGF8FICTyUOuqQ9TVOkcvkUdTJElliFRVIt5vqKdQsedMP1vzAv7tZ6cMScqJo934beOJRCA+myprJ5AvK9BXUMRlATcQYA+T75Jt7vxlFOJzY52aGa7074ZYkNBvV4Asf3WW+mAILLsC7+vWENtIO744u40eJX3C7IpjkM1lC9QVjiTJ9gQog0vVDceXvfZVv+zfxzDK3/Jke4UU8h5CC5wA80Lo0jM3cmsofjJTAg9SJ70iJ44uc4kr2jHRjqJMFsMQqKpCGLTUu4u9TxvfT0BJ/MNFTn8tq8Yw2mUE/F6PwYKqsmaD5J6GCOlbjNnUxbQD+BlbGJ9923Q4VQzh83mpUE4NwSxKcjWoVgF4ardS4WHAB3tPPEzpIO7xjO4z5JyG4Xbna3SuyhOoLbAn+Nlx9A16I2wFYOUb1BZlMkW5uiCt/7yszmliQQ14x2d44+0dD4cEBIaRgehU+/u93wjh8iLMHIzGQ5ZK6EpuvCYwY6iQXllhFBdIj2EpXFsRL3umfe3K+JKWeRmRWPNGCDnpQeDBV1kzgp1UZFdQxGaDk8rr7PkUU4fU9pVU614RbUuK4qHYJ9LxbcgHe03+uo5CvgeCO7TKelWNfRWMYm+OnpEf1BX4VoVxfkEiS9MTiXJcLEO5H1UW9Z4em+vG3ooIjkBRcHcxpXlVoMFACgK+usUmJcpo8gjpZXpUSgd7BVvOUvOSd80tS9WWyB591G4tdrRXOaSoDlFxev42lKzdcOPylAznVdS5LX7ioDsxbxFXJBXhP/2RfIUeg7TIOnCtXtDIcaQmgvqBfRSjZF0RRkvTs3isyih0ZvcU/hSNTFY0MJAXT+0VL9NNfUYkuKuXJVyd5VCkR6J55d0eWdL88XyD1eL5b+xI97v8l9evGeMMduVrV2mEiidBAMoBagPYN0BiMhr/Biw4XXrvKhfxZ97LklY9qw8h7ZqUf5/fxaPc4vqB+foMd9zNhhLKux/ihsjhF8tHTA/UF3xJF+4IoV5JunX1TEQ5+2nbfD4lxHUgqBoUKMb6/1SnDXSZc9bx7iwy7XcWA3ktHB7ie2a57XU0bwzKWKvxxprcn1JSPb4Cmp9DVC3CrH/AUvJ4UP7JKf9EFeF//1yrSr94srAidYaUlVF/wLVG0LyBmFNLeGhJTHbtPdwtcLwywJa3gL3mbvSuTjbTiDaeVtbEXJJcfJ1QaoNzyek2/nhiHypOCMt3HYsstwA/p99Ypo+mZ3IwGNQJoui8iqS/goqM49KsIRfsC5o1D+sOSx/g0toBrgc+Iy0/KMH/Je67nJvWvfsUPqKiNTa36BcW8lgYot7xer5RYTJEjsFrLGFluAX5If30woqCAqFcH5M2fVVqC+sLQqwiF+wJAHNJTa/kFjFM+fSwwtQjPgC84ennJ+w4ndPji635QRW2sop8DxR4N4C9AswGyFqDbO74FnorXrxzSn3cBflj/T1RigKp+DnyyrwS/L7Alsr5TBkN9AWKRfpfPcnw9tgD+H2AD3lsblLFEvT+Uq5I21rh0qJyYAWiAsgvQvxqTx5OGdWVE/bM0vRBZy0cyHvxiFXclROwLsUj/u1jqxjjfVxa4BD57qlS/uGvYDJW0sUYpdsPk0kIuwW9tlVrzb+ylQT6lwifW4umv7VQh/WNVgCZgTqxF7AtHRTpBxzjoaWCBif2SIl9fMt9xy1baANOd4yalHJ9/Wy4bfL/oN/LLWgK+X/qCMcXYN7bA2AJjCzxNLfCv3oPSLTk4TZQAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x^{5} p{\\left (x \\right )} e^{s x}\\, dx - \\frac{5}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right) \\int x^{4} p{\\left (x \\right )} e^{s x}\\, dx - \\frac{10}{\\int p{\\left (x \\right )} e^{s x}\\, dx} \\left(\\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx\\right) \\int x^{3} p{\\left (x \\right )} e^{s x}\\, dx + \\frac{20}{\\left(\\int p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}} \\left(\\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}\\right) \\int x^{3} p{\\left (x \\right )} e^{s x}\\, dx + \\frac{30}{\\left(\\int p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2}} \\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right) \\left(\\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx\\right)^{2} - \\frac{60}{\\left(\\int p{\\left (x \\right )} e^{s x}\\, dx\\right)^{3}} \\left(\\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right)^{3}\\right) \\int x^{2} p{\\left (x \\right )} e^{s x}\\, dx + \\frac{24 \\left(\\int x p{\\left (x \\right )} e^{s x}\\, dx\\right)^{5}}{\\left(\\int p{\\left (x \\right )} e^{s x}\\, dx\\right)^{4}}\\right)$$"
],
"text/plain": [
" \n",
" ⎛⌠ ⎞ ⌠ ⎛⌠ \n",
" ⎜⎮ s⋅x ⎟ ⎮ 4 s⋅x ⎜⎮ 2 s⋅\n",
"⌠ 5⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟⋅⎮ x ⋅p(x)⋅ℯ dx 10⋅⎜⎮ x ⋅p(x)⋅ℯ \n",
"⎮ 5 s⋅x ⎝⌡ ⎠ ⌡ ⎝⌡ \n",
"⎮ x ⋅p(x)⋅ℯ dx - ────────────────────────────────────── - ─────────────────\n",
"⌡ ⌠ ⌠ \n",
" ⎮ s⋅x ⎮ \n",
" ⎮ p(x)⋅ℯ dx ⎮ p(\n",
" ⌡ ⌡ \n",
" \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
" \n",
"\n",
" 2 \n",
" ⎞ ⌠ ⎛⌠ ⎞ ⌠ ⎛⌠ \n",
"x ⎟ ⎮ 3 s⋅x ⎜⎮ s⋅x ⎟ ⎮ 3 s⋅x ⎜⎮ \n",
" dx⎟⋅⎮ x ⋅p(x)⋅ℯ dx 20⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟ ⋅⎮ x ⋅p(x)⋅ℯ dx 30⋅⎜⎮ x⋅p\n",
" ⎠ ⌡ ⎝⌡ ⎠ ⌡ ⎝⌡ \n",
"─────────────────────── + ──────────────────────────────────────── + ─────────\n",
" 2 \n",
" s⋅x ⎛⌠ ⎞ \n",
"x)⋅ℯ dx ⎜⎮ s⋅x ⎟ \n",
" ⎜⎮ p(x)⋅ℯ dx⎟ \n",
" ⎝⌡ ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" ⌠ \n",
" ⎮ s⋅x \n",
" ⎮ p(x)⋅ℯ dx \n",
" ⌡ \n",
"\n",
" 2 3 \n",
" ⎞ ⎛⌠ ⎞ ⎛⌠ ⎞ ⌠ \n",
" s⋅x ⎟ ⎜⎮ 2 s⋅x ⎟ ⎜⎮ s⋅x ⎟ ⎮ 2 s⋅x \n",
"(x)⋅ℯ dx⎟⋅⎜⎮ x ⋅p(x)⋅ℯ dx⎟ 60⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟ ⋅⎮ x ⋅p(x)⋅ℯ dx \n",
" ⎠ ⎝⌡ ⎠ ⎝⌡ ⎠ ⌡ \n",
"───────────────────────────────── - ──────────────────────────────────────── +\n",
" 2 3 \n",
" ⎛⌠ ⎞ ⎛⌠ ⎞ \n",
" ⎜⎮ s⋅x ⎟ ⎜⎮ s⋅x ⎟ \n",
" ⎜⎮ p(x)⋅ℯ dx⎟ ⎜⎮ p(x)⋅ℯ dx⎟ \n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
" \n",
"\n",
" 5\n",
" ⎛⌠ ⎞ \n",
" ⎜⎮ s⋅x ⎟ \n",
" 24⋅⎜⎮ x⋅p(x)⋅ℯ dx⎟ \n",
" ⎝⌡ ⎠ \n",
" ──────────────────────\n",
" 4 \n",
" ⎛⌠ ⎞ \n",
" ⎜⎮ s⋅x ⎟ \n",
" ⎜⎮ p(x)⋅ℯ dx⎟ \n",
" ⎝⌡ ⎠ \n",
"───────────────────────\n",
" \n",
" \n",
" \n",
" "
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant_expression(k=5)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
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\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )}\\, dx} \\left(\\frac{24 \\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{5}}{\\left(\\int p{\\left (x \\right )}\\, dx\\right)^{4}} - \\frac{60}{\\left(\\int p{\\left (x \\right )}\\, dx\\right)^{3}} \\left(\\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{3}\\right) \\int x^{2} p{\\left (x \\right )}\\, dx + \\frac{20}{\\left(\\int p{\\left (x \\right )}\\, dx\\right)^{2}} \\left(\\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{2}\\right) \\int x^{3} p{\\left (x \\right )}\\, dx + \\frac{30}{\\left(\\int p{\\left (x \\right )}\\, dx\\right)^{2}} \\left(\\int x p{\\left (x \\right )}\\, dx\\right) \\left(\\int x^{2} p{\\left (x \\right )}\\, dx\\right)^{2} - \\frac{5}{\\int p{\\left (x \\right )}\\, dx} \\left(\\int x p{\\left (x \\right )}\\, dx\\right) \\int x^{4} p{\\left (x \\right )}\\, dx - \\frac{10}{\\int p{\\left (x \\right )}\\, dx} \\left(\\int x^{2} p{\\left (x \\right )}\\, dx\\right) \\int x^{3} p{\\left (x \\right )}\\, dx + \\int x^{5} p{\\left (x \\right )}\\, dx\\right)$$"
],
"text/plain": [
" \n",
" 5 3 ⌠ 2 ⌠ \n",
" ⎛⌠ ⎞ ⎛⌠ ⎞ ⎮ 2 ⎛⌠ ⎞ ⎮ 3 \n",
"24⋅⎜⎮ x⋅p(x) dx⎟ 60⋅⎜⎮ x⋅p(x) dx⎟ ⋅⎮ x ⋅p(x) dx 20⋅⎜⎮ x⋅p(x) dx⎟ ⋅⎮ x ⋅p(\n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ ⌡ ⎝⌡ ⎠ ⌡ \n",
"───────────────── - ────────────────────────────── + ─────────────────────────\n",
" 4 3 2 \n",
" ⎛⌠ ⎞ ⎛⌠ ⎞ ⎛⌠ ⎞ \n",
" ⎜⎮ p(x) dx⎟ ⎜⎮ p(x) dx⎟ ⎜⎮ p(x) dx⎟ \n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ ⎝⌡ ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" 2 \n",
" ⎛⌠ ⎞ ⌠ ⎛\n",
" ⎛⌠ ⎞ ⎜⎮ 2 ⎟ ⎛⌠ ⎞ ⎮ 4 ⎜\n",
"x) dx 30⋅⎜⎮ x⋅p(x) dx⎟⋅⎜⎮ x ⋅p(x) dx⎟ 5⋅⎜⎮ x⋅p(x) dx⎟⋅⎮ x ⋅p(x) dx 10⋅⎜\n",
" ⎝⌡ ⎠ ⎝⌡ ⎠ ⎝⌡ ⎠ ⌡ ⎝\n",
"───── + ──────────────────────────────── - ──────────────────────────── - ────\n",
" 2 ⌠ \n",
" ⎛⌠ ⎞ ⎮ p(x) dx \n",
" ⎜⎮ p(x) dx⎟ ⌡ \n",
" ⎝⌡ ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" ⌠ \n",
" ⎮ p(x) dx \n",
" ⌡ \n",
"\n",
" \n",
"⌠ ⎞ ⌠ \n",
"⎮ 2 ⎟ ⎮ 3 \n",
"⎮ x ⋅p(x) dx⎟⋅⎮ x ⋅p(x) dx ⌠ \n",
"⌡ ⎠ ⌡ ⎮ 5 \n",
"────────────────────────── + ⎮ x ⋅p(x) dx\n",
" ⌠ ⌡ \n",
" ⎮ p(x) dx \n",
" ⌡ \n",
" \n",
"─────────────────────────────────────────\n",
" \n",
" \n",
" "
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant(k=5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Generating latex expression"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\log{\\left (\\int p{\\left (x \\right )} e^{s x}\\, dx \\right )}$$"
],
"text/plain": [
" ⎛⌠ ⎞\n",
" ⎜⎮ s⋅x ⎟\n",
"log⎜⎮ p(x)⋅ℯ dx⎟\n",
" ⎝⌡ ⎠"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cgf"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"'\\\\log{\\\\left (\\\\int p{\\\\left (x \\\\right )} e^{s x}\\\\, dx \\\\right )}'"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.latex(cgf)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$$\\frac{1}{\\int p{\\left (x \\right )}\\, dx} \\left(- \\frac{\\left(\\int x p{\\left (x \\right )}\\, dx\\right)^{2}}{\\int p{\\left (x \\right )}\\, dx} + \\int x^{2} p{\\left (x \\right )}\\, dx\\right)$$"
],
"text/plain": [
" 2 \n",
" ⎛⌠ ⎞ \n",
" ⎜⎮ x⋅p(x) dx⎟ ⌠ \n",
" ⎝⌡ ⎠ ⎮ 2 \n",
"- ────────────── + ⎮ x ⋅p(x) dx\n",
" ⌠ ⌡ \n",
" ⎮ p(x) dx \n",
" ⌡ \n",
"───────────────────────────────\n",
" ⌠ \n",
" ⎮ p(x) dx \n",
" ⌡ "
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"kth_cumulant(2)"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"'\\\\frac{1}{\\\\int p{\\\\left (x \\\\right )}\\\\, dx} \\\\left(- \\\\frac{\\\\left(\\\\int x p{\\\\left (x \\\\right )}\\\\, dx\\\\right)^{2}}{\\\\int p{\\\\left (x \\\\right )}\\\\, dx} + \\\\int x^{2} p{\\\\left (x \\\\right )}\\\\, dx\\\\right)'"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sympy.latex(kth_cumulant(2))"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.5"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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tanemaki commented Apr 13, 2021

IMG_0747

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