Added tut_06b

tutorials/tut_06.ipynb

@@ -94,7 +94,7 @@
     "\\end{array}\n",
     "\\right)\n",
     "\\end{equation}\n",
-    "The $y$'s in this case are the $D_L$'s.  Now we need to weight by the errors.  There doesn't seem to be a function for doing this automatically in python, but we can do it ourselves.  If the dependent variables and the columns of the M matrix are divided by the standard deviations this will be equivalent to the chi-squared weighting as described in the lecture notes.\n",
+    "The $y$'s in this case are the $D_L$'s.  Now we need to weight by the errors.  There doesn't seem to be a function for doing this automatically in python, but we can do it ourselves.  If the dependent variables and the columns of the M matrix are divided by the standard deviations this will be equivalent to the chi-squared weighting as described in the lecture notes (chapter 6, section 5).\n",
     "\n",
     "Fit a second order polynomial to $D_L$ vs $z$. Plot it and the data."
    ]
@@ -126,7 +126,7 @@
     "\n",
     "$q_o = \\frac{\\Omega_m}{2} - \\Omega_\\Lambda$\n",
     "\n",
-    "where $\\Omega_m$ is the density of the Universe in ordinary matter in units of the critical density and $\\Omega_\\Lambda$ is the energy density of the cosmological constant in the same units.\n",
+    "where $\\Omega_m$ is the density of the Universe in ordinary matter in units of the critical density and $\\Omega_\\Lambda$ is the energy density of the cosmological constant in the same units.  This is an approximation that was used for may years before observations extended out to $z>1$.\n",
     "\n",
     "If the Universe is geometrically flat $\\Omega_m+ \\Omega_\\Lambda = 1$ and \n",
     "\n",
@@ -228,8 +228,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "7) Why might a person object to using a $\\chi^2_{n-2}$ distribution \n",
-    "above in determining the significance of the result?"
+    "~~7) Why might a person object to using a $\\chi^2_{n-2}$ distribution above in determining the significance of the result?~~"
    ]
   },
   {
@@ -251,7 +250,40 @@
    "execution_count": null,
    "metadata": {},
    "outputs": [],
-   "source": []
+   "source": [
+    "from scipy.stats import chi2\n",
+    "qo=np.linspace(...)\n",
+    "muo=np.linspace(...)\n",
+    "x2d,y2d = np.meshgrid(qo,muo)\n",
+    "\n",
+    "print(x2d.shape)\n",
+    "\n",
+    "X2 = np.zeros(x2d.shape)\n",
+    "for i in ... :\n",
+    "    for j in ... :\n",
+    "        X = ...\n",
+    "\n",
+    "        X2[i,j] = np.dot(X,X)\n",
+    "\n",
+    "n = len(mu)\n",
+    "cl68 = ...\n",
+    "cl95 = ...\n",
+    "cl99 = ...\n",
+    "\n",
+    "fig, ax = plt.subplots()\n",
+    "CS = ax.contour(x2d,y2d,X2,[cl68,cl95,cl99])\n",
+    "\n",
+    "fmt = {}\n",
+    "strs = [ '68%', '95%', '99%']\n",
+    "for i,s in zip( CS.levels, strs ):\n",
+    "    fmt[i] = s\n",
+    "\n",
+    "ax.clabel(CS, inline=True, fontsize=10,fmt=fmt)\n",
+    "\n",
+    "plt.xlabel(r'$q_o$')\n",
+    "plt.ylabel(r'$\\mu_o$')\n",
+    "plt.show()"
+   ]
   },
   {
    "cell_type": "markdown",
@@ -263,11 +295,9 @@
     "\n",
     "Make a nice histogram of the $q_o$ vector.\n",
     "\n",
-    "This vector of $q_o$'s is approximately sampled from the true distribution.  Find the mean and variance of this vector.  This is an estimate of the variance in the estimate of $q_o$.\n",
+    "This vector of $q_o$'s is approximately sampled from the true distribution of possible data sets.  Find the mean and variance of this vector.  This is an estimate of the variance in the estimate of $q_o$.\n",
     "\n",
-    "What is the measured value of $q_o$ with errors?\n",
-    "\n",
-    "Assuming the Universe is flat what is the measured value of $\\Omega_m$ with errors?"
+    "What is the measured value of $q_o$ with errors?\n"
    ]
   },
   {
@@ -276,7 +306,19 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "### put loop here\n"
+    "### put loop here\n",
+    "\n",
+    "\n",
+    "### plot here\n",
+    "\n",
+    "print('qo = ', ... ,' +/- ', ... )"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "10) Assuming the Universe is flat what is the measured value of $\\Omega_m$ with errors?"
    ]
   },
   {
@@ -285,11 +327,11 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "### put plot here\n",
+    "### put plot here for flat case\n",
     ".\n",
     ".\n",
     "\n",
-    "print('qo = ', ... ,' +/- ', ... )\n",
+    "\n",
     "print('Omega_m = ', ... ,' +/- ', ... )\n"
    ]
   },