Merge pull request #269 from rummanwaqar/master

Bug fix: Issues in Chap 3 and 4

Author: rlabbe

03-Gaussians.ipynb

@@ -313,7 +313,7 @@
     "\n",
     "If $x$ is continuous we substitute the sum for an integral, like so\n",
     "\n",
-    "$$\\mathbb E[X] = \\int_{a}^b\\, f(x) \\,dx$$\n",
+    "$$\\mathbb E[X] = \\int_{a}^b\\, xf(x) \\,dx$$\n",
     "\n",
     "where $f(x)$ is the probability distribution function of $x$. We won't be using this equation yet, but we will be using it in the next chapter.\n",
     "\n",
@@ -826,7 +826,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This curve is a [*probability density function*](https://en.wikipedia.org/wiki/Probability_density_function) or *pdf* for short. It shows the relative likelihood  for the random variable to take on a value. We can tell from the chart student is somewhat more likely to have a height near 1.8 m than 1.7 m, and far more likely to have a height of 1.9 m vs 1.4 m. Put another way, many students will have a height near 1.8 m, and very few students will have a height of 1.4 m or 1.9 meaters. Finally, notice that the curve is centered over the mean of 1.8 m.\n",
+    "This curve is a [*probability density function*](https://en.wikipedia.org/wiki/Probability_density_function) or *pdf* for short. It shows the relative likelihood  for the random variable to take on a value. We can tell from the chart student is somewhat more likely to have a height near 1.8 m than 1.7 m, and far more likely to have a height of 1.9 m vs 1.4 m. Put another way, many students will have a height near 1.8 m, and very few students will have a height of 1.4 m or 2.2 meters. Finally, notice that the curve is centered over the mean of 1.8 m.\n",
     "\n",
     "> I explain how to plot Gaussians, and much more, in the Notebook *Computing_and_Plotting_PDFs* in the \n",
     "Supporting_Notebooks folder. You can read it online [here](https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Supporting_Notebooks/Computing_and_plotting_PDFs.ipynb) [1].\n",

04-One-Dimensional-Kalman-Filters.ipynb

@@ -460,7 +460,7 @@
     "\n",
     "$$\\begin{aligned}\n",
     "\\mathcal N(\\mu, \\sigma^2) &= \\| prior \\cdot likelihood \\|\\\\\n",
-    "&=\\mathcal{N}(\\bar\\mu, \\bar\\sigma^2)\\cdot \\mathcal{N}(\\mu_z, \\sigma_z^2) \\\\\n",
+    "&= \\| \\mathcal{N}(\\bar\\mu, \\bar\\sigma^2)\\cdot \\mathcal{N}(\\mu_z, \\sigma_z^2) \\|\\\\\n",
     "&= \\mathcal N(\\frac{\\bar\\sigma^2 \\mu_z + \\sigma_z^2 \\bar\\mu}{\\bar\\sigma^2 + \\sigma_z^2},\\frac{\\bar\\sigma^2\\sigma_z^2}{\\bar\\sigma^2 + \\sigma_z^2})\n",
     "\\end{aligned}$$\n",
     "\n",