{ "cells": [ { "cell_type": "markdown", "id": "7a36b325", "metadata": {}, "source": [ "# Probability Theory Review\n", "\n", "\n", "- **[1]** (a) (#) Proof that the \"elementary\" sum rule $p(A) + p(\\bar{A}) = 1$ follows from the (general) sum rule $$p(A+B) = p(A) + p(B) - p(A,B)\\,.$$ \n", " (b) (###) Conversely, derive the general sum rule\n", " $p(A + B) = p(A) + p(B) - p(A,B)$\n", "from the elementary sum rule $p(A) + p(\\bar A) = 1$ and the product rule. Here, you may make use of the (Boolean logic) fact that $A + B = \\overline {\\bar A \\bar B }$. \n", " \n", "\n", "- **[2]** Box 1 contains 8 apples and 4 oranges. Box 2 contains 10 apples and 2 oranges. Boxes are chosen with equal probability. \n", " (a) (#) What is the probability of choosing an apple? \n", " (b) (##) If an apple is chosen, what is the probability that it came from box 1?\n", "\n", "\n", "- **[3]** (###) The inhabitants of an island tell the truth one third of the time. They lie with probability $2/3$. On an occasion, after one of them made a statement, you ask another \"was that statement true?\" and he says \"yes\". What is the probability that the statement was indeed true?\n", "\n", "\n", "- **[4]** (##) A bag contains one ball, known to be either white or black. A white ball is put in, the bag is shaken, and a ball is drawn out, which proves to be white. What is now the chance of drawing a white ball? (Note that the state of the bag, after the operations, is exactly identical to its state before.)\n", "\n", "\n", "- **[5]** A dark bag contains five red balls and seven green ones. \n", " (a) (#) What is the probability of drawing a red ball on the first draw? \n", " (b) (##) Balls are not returned to the bag after each draw. If you know that on the second draw the ball was a green one, what is now the probability of drawing a red ball on the first draw? \n", "\n", "\n", "- **[6]** (#) Is it more correct to speak about the likelihood of a _model_ (or model parameters) than about the likelihood of an _observed data set_. And why? \n", "\n", "\n", "- **[7]** (##) Is a speech signal a 'probabilistic' (random) or a deterministic signal?\n", "\n", " \n", "- **[8]** (##) Proof that, for any distribution of $x$ and $y$ and $z=x+y$\n", "\\begin{align*}\n", " \\mathbb{E}[z] &= \\mathbb{E}[x] + \\mathbb{E}[y] \\\\\n", " \\mathbb{V}[z] &= \\mathbb{V}[x] + \\mathbb{V}[y] + 2\\mathbb{V}[x,y] \n", "\\end{align*}\n", "where $\\mathbb{E}[\\cdot]$, $\\mathbb{V}[\\cdot]$ and $\\mathbb{V}[\\cdot,\\cdot]$ refer to the expectation (mean), variance and covariance operators respectively. You may make use of the more general theorem that the mean and variance of any distribution $p(x)$ is processed by a linear tranformation as\n", " \\begin{align*}\n", "\\mathbb{E}[Ax +b] &= A\\mathbb{E}[x] + b \\\\\n", "\\mathbb{V}[Ax +b] &= A\\,\\mathbb{V}[x]\\,A^T \n", "\\end{align*}\n" ] }, { "cell_type": "code", "execution_count": null, "id": "4f1a0afe", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.5.2", "language": "julia", "name": "julia-1.5" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.5.2" } }, "nbformat": 4, "nbformat_minor": 5 }