329 lines
73 KiB
Plaintext
329 lines
73 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "206bf674",
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"metadata": {},
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"outputs": [],
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"source": [
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"import os\n",
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"import sys\n",
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"\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"import pandas as pd\n",
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"import seaborn as sns\n",
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"\n",
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"sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "612bd02c",
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"metadata": {},
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"source": [
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"# Chapter 3 Example Problems"
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]
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},
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{
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"cell_type": "markdown",
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"id": "60ec2620",
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"metadata": {},
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"source": [
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"## Example 3.4\n",
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"\n",
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"$$ R_Y = \\mathbb{N} $$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 26,
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"id": "f1631c9c",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": 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",
|
|
"text/plain": [
|
|
"<Figure size 640x480 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"p = 0.4\n",
|
|
"y = np.arange(1, 11) # geometric starts at 1\n",
|
|
"pmf = ((1 - p)**(y - 1)) * p\n",
|
|
"\n",
|
|
"plt.stem(y, pmf)\n",
|
|
"plt.title(r\"$P_Y(y) = (1-p)^{y-1}p$, where $p=$\" + f\"{p}\")\n",
|
|
"plt.xlabel(r\"$y$\")\n",
|
|
"plt.ylabel(r\"$P_Y(y)$\")\n",
|
|
"plt.xticks(y)\n",
|
|
"plt.ylim(bottom=0)\n",
|
|
"plt.tight_layout()\n",
|
|
"plt.show()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "f133a554",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Example 3.5\n",
|
|
"\n",
|
|
"$$P_Y(y) = \\begin{cases}\n",
|
|
" (1 - p)^{y-1}p & \\text{if } y \\in \\mathbb{N} \\\\\n",
|
|
" 0 & \\text{otherwise }\n",
|
|
"\\end{cases}$$\n",
|
|
"\n",
|
|
"1. Note that $R_Y = \\mathbb{N}$. So\n",
|
|
"\n",
|
|
"$$\\sum_{y \\in R_Y}P_Y(y)$$\n",
|
|
"> Note that therewas a textbook typo\n",
|
|
"\n",
|
|
"Turns into\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"\\sum_{y \\in \\mathbb{N}}(1 - p)^{y-1}p \\\\\n",
|
|
"p \\cdot \\sum_{y \\in \\mathbb{N}}(1 - p)^{y-1} \\\\\n",
|
|
"p \\cdot ((1 - p)^{1-1} + (1 - p)^{2-1} + (1 - p)^{3-1} + \\dots) \\\\\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"Let $x = 1 - p$, we have\n",
|
|
"\n",
|
|
"$$p \\cdot (1 + x + x^2 + x^3 + \\dots)$$\n",
|
|
"\n",
|
|
"Is $|x| \\lt 1$? Yes proof:\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"0 \\lt p \\lt 1 \\\\\n",
|
|
"-1 \\lt p - 1 \\lt 0 \\\\\n",
|
|
"-1 \\lt x \\lt 0 \\implies |x| \\lt 1\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"So now we have \n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"p \\cdot (\\frac{1}{1-x}) \\\\\n",
|
|
"p \\cdot (\\frac{1}{1-(1 - p)}) \\\\\n",
|
|
"p \\cdot (\\frac{1}{1-(1 - p)}) \\\\\n",
|
|
"p \\cdot (\\frac{1}{p}) \\\\\n",
|
|
"1\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"$\\therefore \\sum_{y \\in R_Y}P_Y(y) = 1$\n",
|
|
"\n",
|
|
"2.\n",
|
|
"\n",
|
|
"If $p = \\frac{1}{2}$ then $P_Y(y)$ simplifies down to\n",
|
|
"\n",
|
|
"$$P_Y(y) = \\begin{cases}\n",
|
|
" \\frac{1}{2^y} & \\text{if } y \\in \\mathbb{N} \\\\\n",
|
|
" 0 & \\text{otherwise }\n",
|
|
"\\end{cases}$$\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"P(2 \\leq Y \\lt 5) &= P(Y = 2) + P(Y = 3) + P(Y = 4) \\\\\n",
|
|
"&= P_Y(2) + P_Y(3) + P_Y(4) \\\\\n",
|
|
"&= \\frac{1}{2^2} + \\frac{1}{2^3} + \\frac{1}{2^4} = \\frac{7}{16}\n",
|
|
"\\end{align*}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "bb2574f0",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Example 3.7\n",
|
|
"\n",
|
|
"Let $X = X_1, X_2, \\dots, X_n$ where $X_i$ are all independent bernoulli random variables.\n",
|
|
"\n",
|
|
"Let $Y = Y_1, Y_2, \\dots, Y_n$ where $Y_i$ are all independent bernoulli random variables.\n",
|
|
"\n",
|
|
"Then\n",
|
|
"$$ Z = X + Y = X_1 + X_2 + \\dots + X_n + Y_1 + Y_2 + \\dots + Y_n $$\n",
|
|
"\n",
|
|
"So $Z \\sim Binomial(n+m, p)$. So the PMF of $Z$ is\n",
|
|
"\n",
|
|
"$$P_X(k) = \\begin{cases} \n",
|
|
"\\binom{n+m}{k}p^k(1-p)^{n+m-k} & \\text{for } k = 1, 2,3, \\dots, n \\\\ \n",
|
|
"0 & \\text{otherwise} \n",
|
|
"\\end{cases}$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "2e7c2c34",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Example 3.10\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 8,
|
|
"id": "eaf6e80f",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"image/png": 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",
|
|
"text/plain": [
|
|
"<Figure size 640x480 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"def pmf(k):\n",
|
|
" return 1 / (2 ** k)\n",
|
|
"\n",
|
|
"x = np.arange(1, 11, 1) # geometric starts at 1\n",
|
|
"y = pmf(x)\n",
|
|
"\n",
|
|
"plt.stem(x, y)\n",
|
|
"plt.title(r\"$P_X(x)$\")\n",
|
|
"plt.xlabel(r\"$x$\")\n",
|
|
"plt.ylabel(r\"$P_X(x)$\")\n",
|
|
"plt.xticks(x)\n",
|
|
"plt.ylim(bottom=0)\n",
|
|
"plt.tight_layout()\n",
|
|
"plt.show()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 9,
|
|
"id": "182321b9",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"image/png": 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",
|
|
"text/plain": [
|
|
"<Figure size 640x480 with 1 Axes>"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"def pmf(k):\n",
|
|
" return 1 / (2 ** k)\n",
|
|
"\n",
|
|
"x = np.arange(1, 11, 1) # geometric starts at 1\n",
|
|
"cdf = np.cumsum(y)\n",
|
|
"\n",
|
|
"plt.step(x, cdf, where='post')\n",
|
|
"plt.title(r\"$F_X(x)$\")\n",
|
|
"plt.xlabel(r\"$x$\")\n",
|
|
"plt.ylabel(r\"$F_X(x)$\")\n",
|
|
"plt.xticks(x)\n",
|
|
"plt.ylim(0, 1.05)\n",
|
|
"plt.tight_layout()\n",
|
|
"plt.show()"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"id": "d5f8d31c",
|
|
"metadata": {},
|
|
"source": [
|
|
"3.12 Let $X \\sim Bernoulli(p)$. Find $E[X]$.\n",
|
|
"\n",
|
|
"Notice $R_X = \\{1, 0\\}$.\n",
|
|
"\n",
|
|
"$$E[X] = 1 \\cdot p + 0 \\cdot (1 - p) = p$$\n",
|
|
"\n",
|
|
"3.13 Let $X \\sim Geometric(p)$. Find $E[X]$.\n",
|
|
"\n",
|
|
"Notice $R_X = \\mathbb{N}$.\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"E[X] &= 1 \\cdot p(1 - p)^{1 - 1} + 2 \\cdot p(1 - p)^{2 - 1} + 3 \\cdot p(1 - p)^{3 - 1} + \\dots \\\\\n",
|
|
"&= 1p + 2p(1 - p) + 3p(1 - p)^{2} + 4p(1 - p)^{3} + \\dots \\\\\n",
|
|
"&= p \\cdot 1(1 - p)^0 + p \\cdot 2(1 - p)^1 + p \\cdot 3(1 - p)^{2} + p \\cdot 4(1 - p)^{3} + \\dots \\\\\n",
|
|
"&= \\frac{1}{p}\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"3.13 Let $X \\sim Poisson(\\lambda)$. Find $E[X]$.\n",
|
|
"\n",
|
|
"\n",
|
|
"Notice $R_X = \\mathbb{N} \\cup \\{0\\}$.\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"E[X] &= 0 \\cdot \\frac{e^{-\\lambda}\\lambda^0}{0!} + 1 \\cdot \\frac{e^{-\\lambda}\\lambda^1}{1!} + 2 \\cdot \\frac{e^{-\\lambda}\\lambda^2}{2!} + 3 \\cdot \\frac{e^{-\\lambda}\\lambda^3}{3!} + \\cdots \\\\\n",
|
|
"&= \\sum_{x = 0}^\\infty x\\cdot \\frac{e^{-\\lambda}\\lambda^x}{x!} \\\\\n",
|
|
"&= e^{-\\lambda} \\cdot \\sum_{x = 0}^\\infty \\frac{\\lambda^x}{(x-1)!} \\\\\n",
|
|
"\\end{align*}\n",
|
|
"\n",
|
|
"Let $k = x - 1$\n",
|
|
"\n",
|
|
"\n",
|
|
"\\begin{align*}\n",
|
|
"E[X] &= e^{-\\lambda} \\cdot \\sum_{k = 0}^\\infty \\frac{\\lambda^{k+1}}{k!} \\\\\n",
|
|
"&= \\lambda e^{-\\lambda} \\cdot \\sum_{k = 0}^\\infty \\frac{\\lambda^k}{k!} \\\\\n",
|
|
"&= \\lambda e^{-\\lambda} \\cdot e^\\lambda = \\lambda\n",
|
|
"\\end{align*}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 17,
|
|
"id": "b1d97b95",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"V=2.916666666666666, SD=np.float64(1.707825127659933)\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"E = 3.5\n",
|
|
"E2 = E**2\n",
|
|
"\n",
|
|
"V = 0\n",
|
|
"\n",
|
|
"for x in range(1, 7):\n",
|
|
" V += (x**2) * (1/6)\n",
|
|
"\n",
|
|
"V = V - E2\n",
|
|
"SD = np.sqrt(V)\n",
|
|
"\n",
|
|
"print(f\"{V=}, {SD=}\")\n"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "roadmap (3.14.5)",
|
|
"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": "3.14.5"
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 5
|
|
}
|