{ "cells": [ { "cell_type": "markdown", "id": "71655f70", "metadata": {}, "source": [ "# Overview of Probability and Random Variables" ] }, { "cell_type": "markdown", "id": "a1b8b4c8", "metadata": {}, "source": [ "This function adjusts matplotlib settings for a uniform feel in the textbook. Note that with usetex=True, fonts are rendered with LaTeX. This may result in an error if LaTeX is not installed on your system. In that case, you can set usetex to False." ] }, { "cell_type": "code", "execution_count": 1, "id": "4c48b283", "metadata": {}, "outputs": [], "source": [ "if \"setup_text_plots\" not in globals():\n", " from astroML.plotting import setup_text_plots\n", "setup_text_plots(fontsize=8, usetex=False)" ] }, { "cell_type": "markdown", "id": "2823b358", "metadata": {}, "source": [ "## Probability axioms" ] }, { "cell_type": "markdown", "id": "f224fef1", "metadata": {}, "source": [ "Given an event $A$, such as the outcome of a coin toss, we assign it a real number $p(A)$, called the probability of $A$. Note that $p(A)$ could also correspond to a probability that a value of $x$ falls in a $dx$ wide interval around $x$. To qualify as a probability, $p(A)$ must satisfy three Kolmogorov axioms:" ] }, { "cell_type": "markdown", "id": "9179164d", "metadata": {}, "source": [ "1. $p(A) \\geq 0$ for each $A$.\n", "2. $p(\\Omega) = 1$, where $\\Omega$ is the set of all possible outcomes.\n", "3. If $A_1$, $A_2$, . . . are disjoint events, then $p (\\bigcup^{\\infty}_{i=1}A_i) = \\sum_{i=1}^{\\infty}p(A_i)$ where $\\bigcup$ stands for “union.”\n", "\n", "Several useful rules can be derived as a consequence of these axioms." ] }, { "cell_type": "markdown", "id": "3c067a63", "metadata": {}, "source": [ "**Sum rule**: The probability that the union of two events, $A$ and $B$, will happen is given by,\n", "\n", "$$\\qquad\\qquad p(A \\cup B) = p(A) + p(B) - p(A \\cap B)\\qquad\\qquad\\qquad(1)$$\n", "\n", "\n", "which is the sum of $A$ and $B$'s respective probabilities minus the probability that both $A$ and $B$ will happen. The union of two events is the probability that *either* event occurs. The $\\cap$ in the equation stands for \"intersection\", and subtracting the last term, $p(A \\cap B)$, avoids double counting the places that $A$ and $B$ overlap. In the figure below, we show this in a diagram." ] }, { "cell_type": "code", "execution_count": 2, "id": "dc8028a2", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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