{ "cells": [ { "cell_type": "markdown", "id": "8eec6aab", "metadata": {}, "source": [ "---\n", "title: \"Understanding Law of Large Numbers and Central Limit Theorem\"\n", "author: Rafiq Islam\n", "date: \"03/15/2025\"\n", "categories: [Statistics, Probability]\n", "citation: true\n", "search: true\n", "image: lln.png\n", "lightbox: true\n", "listing: \n", " contents: \"/../../posts\"\n", " max-items: 3\n", " type: grid\n", " categories: true\n", " date-format: full\n", " fields: [image, date, title, author, reading-time]\n", "format: \n", " html: default\n", " ipynb: default\n", " docx: \n", " toc: true\n", " adsense:\n", " enable-ads: false\n", " epub:\n", " toc: true\n", " adsense:\n", " enable-ads: false\n", " pdf: \n", " toc: true\n", " pdf-engine: xelatex\n", " adsense:\n", " enable-ads: false\n", " number-sections: false\n", " colorlinks: true\n", "toc-depth: 4\n", "--- \n", "\n", "

\n", "Statistical inference is a fundamental part of data science and machine learning. Two of the most important theorems in probability theory that form the backbone of many statistical methods are the **Law of Large Numbers (LLN)** and the **Central Limit Theorem (CLT)**. It's not uncommon to mix-up with these two concepts often.

\n", "\n", "## The Law of Large Numbers (LLN)\n", "\n", "### Intuition Behind LLN \n", "\n", "

\n", "The Law of Large Numbers states that as the sample size increases, the sample mean approaches the population mean. In other words, with more observations, the average of the sample becomes a better estimate of the true average of the population.

\n", "\n", "### Mathematical Definition\n", "\n", "Let $X_1, X_2, X_3, \\dots, X_n$ be a sequence of independent and identically distributed (i.i.d.) random variables with an expected value $\\mu$. The sample mean is given by:\n", "\n", "$$\n", "\\bar{X_n} = \\frac{1}{n} \\sum_{i=1}^{n} X_i\n", "$$\n", "According to LLN:\n", "$$\n", "\\bar{X_n} \\to \\mu \\quad \\text{as } n \\to \\infty\n", "$$ \n", "\n", "This means that for a sufficiently large $n$, $\\bar{X_n}$ will be very close to $\\mu$. That is \n", "\n", "$$\n", "\\lim_{n\\to \\infty} \\bar{X}_n = \\lim_{n\\to \\infty} \\frac{1}{n} \\sum_{i=1}^{n} X_i = \\mu\n", "$$ \n", "\n", "### Visualization \n", "\n", "Say, we have a population of 150,000 male in a country called VSA (a hypothetical country) " ] }, { "cell_type": "code", "execution_count": 1, "id": "e10b6908", "metadata": {}, "outputs": [], "source": [ "#| code-fold: true\n", "import numpy as np\n", "\n", "heights = np.random.randint(low=90, high=190, size=150000)\n", "mean_height = np.mean(heights)" ] }, { "cell_type": "markdown", "id": "8e4a186e", "metadata": { "user_expressions": [ { "expression": "np.round(mean_height,2)", "result": { "data": { "text/plain": "np.float64(139.52)" }, "metadata": {}, "status": "ok" } } ] }, "source": [ "and the true mean/average height of men is `{python} np.round(mean_height,2)`. We want to see how varying the sample size affect the sample mean and variances. " ] }, { "cell_type": "code", "execution_count": 2, "id": "f079a898", "metadata": {}, "outputs": [ { "data": { "application/pdf": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#| code-fold: true\n", "import matplotlib.pyplot as plt\n", "from mywebstyle import plot_style\n", "plot_style('#f4f4f4')\n", "\n", "sample_sizes = [50, 100, 1000, 5000]\n", "number_of_studies = 10000\n", "# Create subplots\n", "fig, axes = plt.subplots(2, 2, figsize=(8.3, 5.5))\n", "axes = axes.ravel()\n", "\n", "for i, n in enumerate(sample_sizes):\n", " sample_means = np.zeros(number_of_studies)\n", " for j in range(number_of_studies):\n", " sample = np.random.choice(heights, size=n, replace=True)\n", " sample_means[j] = np.mean(sample)\n", " \n", " # Plot histogram\n", " axes[i].hist(sample_means, bins=35, density=True, alpha=0.7, color='blue')\n", " ymin, ymax = axes[i].get_ylim()\n", " axes[i].plot([mean_height, mean_height], [ymin, ymax], color='red', linestyle='-',linewidth=3, label='True Mean')\n", " axes[i].set_title(f\"Sample Size: {n}\")\n", " axes[i].set_xlabel(\"Sample Mean\")\n", " axes[i].set_ylabel(\"Density\")\n", " axes[i].legend()\n", "\n", "plt.tight_layout()\n", "plt.savefig('lln.png')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "6772c23b", "metadata": { "user_expressions": [ { "expression": "np.round(mean_height,2)", "result": { "data": { "text/plain": "np.float64(139.52)" }, "metadata": {}, "status": "ok" } } ] }, "source": [ "

\n", "From the top-left figure, when the sample size is 50, the sample mean varies from 130 to 150 where the true population mean is `{python} np.round(mean_height,2)`, similarly in the 4th figure (bottom-left), when the sample size is 5000, the mean varies from 138 to 141. Thus, if we increase the sample size, sufficiently large, then the sample mean is approximately equal to the population mean. This is what LLN is!

\n", "\n", "\n", "\n", "\n", "### Types of LLN\n", "There are two forms of the Law of Large Numbers:\n", "\n", "1. **Weak Law of Large Numbers (WLLN)**: Convergence of the sample mean to the population mean happens in probability.\n", "2. **Strong Law of Large Numbers (SLLN)**: Convergence happens almost surely, meaning with probability 1.\n", "\n", "\n", "### Why is LLN Important?\n", "LLN justifies the use of sample means in estimation problems. For example, if we want to estimate the average income of a country, we don’t need to survey the entire population; we can take a large enough sample, and the sample mean will approximate the true mean.\n", "\n", "---\n", "\n", "## The Central Limit Theorem (CLT)\n", "\n", "### Intuition Behind CLT \n", "\n", "

\n", "While the Law of Large Numbers tells us that sample means converge to the population mean, the Central Limit Theorem goes further. It states that the distribution of the sample mean follows a normal distribution, regardless of the shape of the population distribution, provided the sample size is large enough.

\n", "\n", "### Mathematical Definition \n", "\n", "Let $X_1, X_2, \\dots, X_n$ be i.i.d. random variables with mean $\\mu$ and variance $\\sigma^2$. Define the sample mean:\n", "$$\n", "\\bar{X_n} = \\frac{1}{n} \\sum_{i=1}^{n} X_i\n", "$$ \n", "\n", "Then, as $n$ increases, the distribution of $\\bar{X_n}$ approaches a normal distribution:\n", "$$\n", "\\frac{\\bar{X_n} - \\mu}{\\sigma / \\sqrt{n}} \\approx N(0,1)\n", "$$ \n", "\n", "This means that if we standardize $\\bar{X_n}$, it follows a standard normal distribution (mean 0, variance 1) when $n$ is large. \n", "\n", "### Visualization " ] }, { "cell_type": "code", "execution_count": 3, "id": "b91a1cb5", "metadata": {}, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#| code-fold: true\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "\n", "# Set random seed for reproducibility\n", "np.random.seed(42)\n", "\n", "# Define population distributions\n", "def generate_population(dist_type, size=1000000):\n", " if dist_type == \"uniform\":\n", " return np.random.uniform(0, 100, size)\n", " elif dist_type == \"exponential\":\n", " return np.random.exponential(scale=10, size=size)\n", " elif dist_type == \"binomial\":\n", " return np.random.binomial(n=10, p=0.5, size=size)\n", " else:\n", " raise ValueError(\"Unknown distribution type\")\n", "\n", "# Function to simulate sample means\n", "def sample_means_experiment(population, sample_size, num_samples=10000):\n", " sample_means = np.zeros(num_samples)\n", " for i in range(num_samples):\n", " sample = np.random.choice(population, size=sample_size, replace=True)\n", " sample_means[i] = np.mean(sample)\n", " return sample_means\n", "\n", "# Define sample sizes\n", "sample_sizes = [5, 100, 1000]\n", "\n", "# Define distributions to test\n", "distributions = [\"uniform\", \"exponential\", \"binomial\"]\n", "\n", "# Plot CLT effect for different distributions\n", "fig, axes = plt.subplots(len(distributions), len(sample_sizes), figsize=(8, 7.5))\n", "fig.suptitle(\"Central Limit Theorem: Sample Mean Distributions\", fontsize=16, fontweight='bold')\n", "\n", "for i, dist_type in enumerate(distributions):\n", " population = generate_population(dist_type)\n", " true_mean = np.mean(population)\n", " \n", " for j, n in enumerate(sample_sizes):\n", " sample_means = sample_means_experiment(population, n)\n", "\n", " # Plot histogram of sample means\n", " sns.histplot(sample_means, bins=20, kde=True, ax=axes[i, j], color='blue')\n", " ymin, ymax = axes[i, j].get_ylim()\n", " axes[i, j].plot([true_mean, true_mean], [ymin, ymax], 'r-', linewidth=2, label=\"True Mean\")\n", " \n", " axes[i, j].set_title(f\"{dist_type.capitalize()} Dist, n = {n}\")\n", " axes[i, j].set_xlabel(\"Sample Mean\")\n", " axes[i, j].set_ylabel(\"Density\")\n", "\n", "plt.tight_layout(rect=[0, 0, 1, 0.96]) # Adjust layout for title\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "cea00b10", "metadata": {}, "source": [ "### Why is CLT Important?\n", "1. **Allows Statistical Inference**: Even if the population distribution is unknown or non-normal, we can still use normal-based statistical techniques when working with large samples.\n", "2. **Enables Hypothesis Testing and Confidence Intervals**: Many inferential statistics methods assume normality. CLT ensures that these methods work for large enough samples.\n", "3. **Makes Sampling Practical**: Without CLT, we would need to know the entire population distribution to make inferences.\n", "\n", "### Example\n", "\n", "

\n", "Suppose we have a population where the income distribution is highly skewed. If we take small samples, their distributions may also be skewed. However, as the sample size increases (e.g., $n > 30$), the distribution of sample means will look more like a normal distribution, allowing us to apply normal-based statistical methods.

\n", "\n", "---\n", "\n", "## Relationship Between LLN and CLT\n", "While both the Law of Large Numbers and the Central Limit Theorem deal with large samples, they serve different purposes:\n", "\n", "- **LLN** guarantees that the sample mean converges to the true population mean as the sample size increases.\n", "- **CLT** ensures that the distribution of the sample mean follows a normal distribution when the sample size is sufficiently large.\n", "\n", "In short, LLN helps us estimate population parameters accurately, while CLT helps us conduct statistical inference using normal approximations.\n", "\n", "---\n", "\n", "## Conclusion \n", "

\n", "The **Law of Large Numbers** and the **Central Limit Theorem** are two of the most fundamental theorems in probability and statistics. LLN reassures us that as we collect more data, our sample mean becomes a reliable estimate of the population mean. CLT, on the other hand, enables powerful statistical techniques by ensuring that sample means follow a normal distribution, even when the underlying population is not normal.\n", "

\n", "Understanding these concepts is essential for data science, as they form the basis for many machine learning and statistical inference methods. Whether you are estimating a population parameter, conducting hypothesis tests, or building predictive models, LLN and CLT provide the theoretical foundation for making reliable decisions based on data.

\n", "\n", "---\n", "\n", "### Further Reading\n", "- \"Introduction to Probability\" by Joseph K. Blitzstein and Jessica Hwang\n", "- \"The Elements of Statistical Learning\" by Hastie, Tibshirani, and Friedman\n", "- Any introductory statistics textbook covering probability theory\n", "\n", "\n", "**Share on** \n", "\n", "::::{.columns}\n", ":::{.column width=\"33%\"}\n", "\n", " \n", "{{< fa brands facebook size=3x >}}\n", "\n", " \n", ":::\n", " \n", ":::{.column width=\"33%\"}\n", "\n", " \n", "{{< fa brands linkedin size=3x >}}\n", "\n", " \n", ":::\n", " \n", ":::{.column width=\"33%\"}\n", "\n", " \n", "{{< fa brands twitter size=3x >}}\n", "\n", " \n", ":::\n", "::::\n", " \n", "\n", " \n", "
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