![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\"){kind=icon}
"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "S1k2jkcdPPB-"
},
"source": [
"## Question 1a (10 points)\n",
"\n",
"In a private subreddit people are posting their opinions on the CEO of the company you work for. Lets assume that the employees that are posting are random logging in to that subreddit and that each post indicates whether the employee approves or not the job that the CEO is doing. Let $x_i$ be the binary random variable where $x_i=1$ indicates approval. You can assume that $x$ is distributed according to a Bernoulli distribution with parameter $p=1/2$.\n",
"\n",
"Your job is to sample $n=50$ posts and estimate the approval rate of the CEO by considering the statistics of $y=x_1+x_2+ \\dots + x_n$. What is the probability that 25 employees approve the CEO?\n",
"\n",
"## Question 1b (10 points)\n",
"\n",
"Following your findings in Q1a, read about the [Central Limit Theorem](https://en.wikipedia.org/wiki/Central_limit_theorem) and recognize that \n",
"\n",
"$$z=\\frac{y-\\mu_y}{\\sigma_y}$$ \n",
"\n",
"is normally distributed with mean 0 and variance 1.\n",
"\n",
"Can you find the probability that 25 employees approve the CEO using the Gaussian approximation?\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "XlTvVPAxPPCF"
},
"source": [
"### 1a\n",
"$P(x) = \\begin{pmatrix}n \\\\ x \\end{pmatrix} p^x (1 - p)^{n - x}$\n",
"\n",
"$n = 50 \\\\ p = 1/5 \\\\ x = 25$\n",
"\n",
"$P(25) = \\begin{pmatrix}50 \\\\ 25 \\end{pmatrix} (1/2)^{25} (1 - 1/2)^{50 - 25}$\n",
"\n",
"$ = {50! \\over 25!(50-25)!} (1/2)^{25} (1 - 1/2)^{50 - 25}$\n",
"\n",
"$ = 0.1123$\n"
]
},
{
"cell_type": "markdown",
"source": [
"### 1b\n",
"$\\mu_y = np = 50 (1/2) = 25$\n",
"\n",
"$\\sigma_y = \\sqrt{np(1-p)} = \\sqrt{50(1/2)(1/2)} = 3.5355$\n",
"\n",
"$Y \\sim \\mathcal{N}(25, 3.5355)$\n",
"\n",
"\n",
"$ P(X=25) = P(24.5 < Y < 25.5)$\n",
"\n",
"$ = 0.1125 $\n"
],
"metadata": {
"id": "KM4GMZ2tPU5x"
}
},
{
"cell_type": "markdown",
"metadata": {
"id": "pl3VkaUXPPCG"
},
"source": [
"## Question 2 (20 points)\n",
"\n",
"A sequential experiment involves repeatedly drawing a ball from one of the two urns, noting the number on the ball and replacing the ball in the urn. Urn 0 contains a ball with the number 0 and two balls with the number 1. Urn 1 contains five balls with the number 0 and one ball with the number 1. \n",
"\n",
"The urn from which the first ball is drawn is selected by flipping a fair coin. Urn 0 is used if the outcome is H and urn 1 is used if the outcome is T. **The urn used in a subsequent draws corresponds to the number on the ball drawn in the previous draw.** \n",
"\n",
"What is the probability of a specific sequence of the numbers on drawn balls being 0011 ? \n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "oGlxDQSvPPCH"
},
"source": [
"Start with coin flip H:\n",
"- 0: Drawing from urn 0, 0 is 1/3\n",
"- 0: Drawing from urn 0, 0 is 1/3\n",
"- 1: Drawing from urn 0, 1 is 2/3\n",
"- 1: Drawing from urn 1, 1 is 1/6\n",
"\n",
"Start with coin flip T:\n",
"- 0: Drawing from urn 1, 0 is 5/6\n",
"- 0: Drawing from urn 0, 0 is 1/3\n",
"- 1: Drawing from urn 0, 1 is 2/3\n",
"- 1: Drawing from urn 1, 1 is 1/6\n",
"\n",
"\n",
"$ = (1/2)((1/3 * 1/3 * 2/3 * 1/6) + (5/6 * 1/3 * 2/3 * 1/6))$\n",
"\n",
"$ = 0.0216$"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "Lhe8p7TXPPCH"
},
"source": [
"## Question 3 (20 points) \n",
"\n",
"Referring to Example 6.6 of the [Math for ML book](https://mml-book.github.io/book/mml-book.pdf), simulate and plot the bivariate normal distribution with the shown parameters using the [Cholesky factorization](https://numpy.org/doc/stable/reference/generated/numpy.linalg.cholesky.html) for the simulation. \n"
]
},
{
"cell_type": "markdown",
"source": [
"$p(x_1, x_2) = \\mathcal{N} = (\\begin{bmatrix}0 \\\\ 2 \\end{bmatrix}, \\begin{bmatrix} 0.3 & -1 \\\\ -1 & 5 \\end{bmatrix})$\n",
"\n",
"$y = Ax + \\mu$"
],
"metadata": {
"id": "DOn70il0YjOk"
}
},
{
"cell_type": "code",
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import seaborn as sns\n",
"import matplotlib.pyplot as plt\n",
"\n",
"np.random.seed(0)\n",
"sns.set_theme(style='whitegrid', palette='pastel')\n",
"\n",
"import warnings\n",
"warnings.filterwarnings('ignore')"
],
"metadata": {
"id": "WVPGNNzFbOkG"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "code",
"source": [
"mean = [0, 2]\n",
"cov = [[0.3, -1], [-1, 5]]\n",
"\n",
"L = np.linalg.cholesky(cov)\n",
"\n",
"samples = np.random.normal(size=(1000, 2))\n",
"\n",
"samples = np.dot(samples, L.T) + mean\n",
"\n",
"sns.scatterplot(x=samples[:, 0], y=samples[:, 1])\n",
"\n",
"plt.show()"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 268
},
"id": "AvhnGPkUYHCj",
"outputId": "123429ad-8292-41f4-de61-d01c789bef91"
},
"execution_count": null,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"| \n", " | Unnamed: 0 | \n", "Date | \n", "Day | \n", "High Temp (°F) | \n", "Low Temp (°F) | \n", "Precipitation | \n", "Brooklyn Bridge | \n", "Manhattan Bridge | \n", "Williamsburg Bridge | \n", "Queensboro Bridge | \n", "Total | \n", "
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | \n", "0 | \n", "2016-04-01 00:00:00 | \n", "2016-04-01 00:00:00 | \n", "78.1 | \n", "66.0 | \n", "0.01 | \n", "1704.0 | \n", "3126 | \n", "4115.0 | \n", "2552.0 | \n", "11497 | \n", "
| 1 | \n", "1 | \n", "2016-04-02 00:00:00 | \n", "2016-04-02 00:00:00 | \n", "55.0 | \n", "48.9 | \n", "0.15 | \n", "827.0 | \n", "1646 | \n", "2565.0 | \n", "1884.0 | \n", "6922 | \n", "
| 2 | \n", "2 | \n", "2016-04-03 00:00:00 | \n", "2016-04-03 00:00:00 | \n", "39.9 | \n", "34.0 | \n", "0.09 | \n", "526.0 | \n", "1232 | \n", "1695.0 | \n", "1306.0 | \n", "4759 | \n", "
| 3 | \n", "3 | \n", "2016-04-04 00:00:00 | \n", "2016-04-04 00:00:00 | \n", "44.1 | \n", "33.1 | \n", "0.47 (S) | \n", "521.0 | \n", "1067 | \n", "1440.0 | \n", "1307.0 | \n", "4335 | \n", "
| 4 | \n", "4 | \n", "2016-04-05 00:00:00 | \n", "2016-04-05 00:00:00 | \n", "42.1 | \n", "26.1 | \n", "0 | \n", "1416.0 | \n", "2617 | \n", "3081.0 | \n", "2357.0 | \n", "9471 | \n", "