Add key equations

benmarlin · benmarlin · commit 267be4e5b891 · 2024-09-15T15:03:52.000-04:00
diff --git a/Data_Generating_Distributions.ipynb b/Data_Generating_Distributions.ipynb
@@ -134,7 +134,10 @@
     "id": "o-fcWbHQQ7i1"
    },
    "source": [
-    "## Data Generating Process Visualization"
+    "## Data Generating Process Visualization\n",
+    "We visualize a joint density $p_*(X=x,Y=y)$ where the density on X is uniform on [0,1] and the density on Y is a conditional Gaussian where the mean depends linearly on X. This yields the joint distribution:\n",
+    "\n",
+    "$p_*(X=x,Y=y) = p_*(X=x)p_*(Y=y|X=x) = U(x;0,1)\\mathcal{N}(y;xw+b,\\sigma^2)$"
    ]
   },
   {

Original file line number	Diff line number	Diff line change
`@@ -134,7 +134,10 @@`
`134`	`134`	`"id": "o-fcWbHQQ7i1"`
`135`	`135`	`},`
`136`	`136`	`"source": [`
`137`		`- "## Data Generating Process Visualization"`
	`137`	`+ "## Data Generating Process Visualization\n",`
	`138`	`+ "We visualize a joint density $p_*(X=x,Y=y)$ where the density on X is uniform on [0,1] and the density on Y is a conditional Gaussian where the mean depends linearly on X. This yields the joint distribution:\n",`
	`139`	`+ "\n",`
	`140`	`+ "$p_(X=x,Y=y) = p_(X=x)p_*(Y=y\|X=x) = U(x;0,1)\\mathcal{N}(y;xw+b,\\sigma^2)$"`
`138`	`141`	`]`
`139`	`142`	`},`
`140`	`143`	`{`