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    "# 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]** (##) <a id=\"distribution-of-sum\">Proof</a> 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"
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