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   "source": [
    "## Exercise 4.1\n",
    "Prove that the Python function `gauss_sample` defined in Chapter 4 indeed generates samples of a Gaussian distribution with mean $c$ and covariance $\\Sigma$."
   ]
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   "source": [
    "## Exercise 4.2\n",
    "\n",
    "Recall the definition of a covariance matrix. Implement a version of the NumPy function `np.cov` yourself (you may call it `my_cov`, for example). Then go to the NumPy documentation of `np.cov` (<https://numpy.org/doc/stable/reference/generated/numpy.cov.html>) and click on the `[source]` link of that page. After some searching, you should be able to view the Python implementation of `np.cov`. Can you identify the part where the actual computation happens?"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4.3\n",
    "\n",
    "Recall Example 4.1 from Chapter 4 that produces a contour plot of the `gauss_density`. \n",
    "\n",
    "That code uses a nested for-loop over all entries in the matrices `X0` and `X1` to compute the corresponding entry in `Z`. This is inefficient for large matrices for at least two reasons:\n",
    "\n",
    "1.  every call of `gauss_density` computes the same normalisation constant anew, even though this does not depend on `x`\n",
    "2.  the numerical libraries underlying NumPy can perform vector and matrix operations much more efficiently than element-wise operations. Rewriting code to make best use of vector and matrix operations is called **vectorization** or [array programming](https://en.wikipedia.org/wiki/Array_programming).\n",
    "\n",
    "Write a function `gauss_density_2d(X0, X1, c, Sigma)` that accepts NumPy matrices `X0, X1` of 2D point coordinates and returns `Z` in one go. Can you avoid using any for-loops? Perform a timing comparison of both approaches for evaluating the Gaussian density function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4.4\n",
    "\n",
    "What is the interpretation of the empirical error $\\hat R(h)\\approx 0.0016$ in Example 4.2, in terms of individual data points?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4.5\n",
    "\n",
    "Try increasing `N` in the code of Example 4.2. You'll notice that the empirical error converges to a value $\\approx 0.002112$. Write down a formula for the precise limiting value."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4.6\n",
    "\n",
    "In Section 4.5 we have implemented the Bayes hypothesis `h_best_L2` and computed its generalisation error. As discussed there, `h_best_L2` does not necessarily attain integer values in $\\{0,1,2\\}$. Can you come up with a best possible hypothesis $h$ that takes only values in $\\{0,1,2\\}$? And compute its generalisation error?"
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