From 4cf8cfd0793fedbff90dbcc73a069cba13c802f4 Mon Sep 17 00:00:00 2001
From: Caleb Burke <calebburke91@gmail.com>
Date: Sat, 30 May 2026 00:14:44 -0700
Subject: [PATCH] small chapter 3 prog

---
 .../ch3/notes.ipynb                           | 50 ++++++++++++++++++-
 1 file changed, 49 insertions(+), 1 deletion(-)

diff --git a/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb b/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb
index 58c31e2..6146982 100644
--- a/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb
+++ b/study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb
@@ -31,7 +31,7 @@
    "id": "9f0046c2",
    "metadata": {},
    "source": [
-    "## Random Variables (3.1.1 - 3.1.2)"
+    "## Random Variables (3.1.1 - 3.1.3)"
    ]
   },
   {
@@ -99,6 +99,54 @@
    "cell_type": "markdown",
    "id": "57323d97",
    "metadata": {},
+   "source": [
+    "## Independent Random Variables (3.1.4)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c6261f7d",
+   "metadata": {},
+   "source": [
+    "***Definition.*** Consider two discrete random variables $X$ and $Y$. We say that $X$ and $Y$ are independent if:\n",
+    "\n",
+    "$$P(X=x, Y=y) = P(X=x)P(Y=y)$$\n",
+    "\n",
+    "In general, if two random variables are independent, then you can write\n",
+    "\n",
+    "$$P(X \\in A, Y \\in B) = P(X \\in A)(Y \\in B)$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b0362c1",
+   "metadata": {},
+   "source": [
+    "***Definition.*** Consider $n$ discrete random variables $X_1, X_2, X_3, \\dots, X_n$. We say that $X_1, X_2, X_3, \\dots, X_n$ are independent if:\n",
+    "\n",
+    "$$P(X_1 = x_1, X_2 = x_2, X_3 = x_3, \\dots, X_n = x_n) = P(X_1 = x_1)P(X_2 = x_2) \\dots P(X_n = x_n)$$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e50c7c48",
+   "metadata": {},
+   "source": [
+    "## Special Distributions (3.1.5)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "853e9acb",
+   "metadata": {},
+   "source": [
+    "***Definition.*** A random variable $X$ is said to be a *Bernoulli* random variable with parameter *p*, shown as $X \\sim \\text{Bernoulli}(p),$ if its PMF is given by"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9e1caee1",
+   "metadata": {},
    "source": []
   }
  ],