big progress on chapter 3

study/001_introduction-to-probability-statistics-and-random-processes/README.md

@@ -4,7 +4,7 @@
 
 total pages=1007
 
-**Currently reading:** chapter 3, page 162
+**Currently reading:** chapter 3, page 190
 
 TODO:
 

study/001_introduction-to-probability-statistics-and-random-processes/ch3/example-problems.ipynb

@@ -137,7 +137,23 @@
    "cell_type": "markdown",
    "id": "bb2574f0",
    "metadata": {},
-   "source": []
+   "source": [
+    "## Example 3.7\n",
+    "\n",
+    "Let $X = X_1, X_2, \\dots, X_n$ where $X_i$ are all independent bernoulli random variables.\n",
+    "\n",
+    "Let $Y = Y_1, Y_2, \\dots, Y_n$ where $Y_i$ are all independent bernoulli random variables.\n",
+    "\n",
+    "Then\n",
+    "$$ Z = X + Y = X_1 + X_2 + \\dots + X_n + Y_1 + Y_2 + \\dots + Y_n $$\n",
+    "\n",
+    "So $Z \\sim Binomial(n+m, p)$. So the PMF of $Z$ is\n",
+    "\n",
+    "$$P_X(k) = \\begin{cases} \n",
+    "\\binom{n+m}{k}p^k(1-p)^{n+m-k}     & \\text{for } k = 1, 2,3, \\dots, n \\\\  \n",
+    "0     & \\text{otherwise} \n",
+    "\\end{cases}$$"
+   ]
   }
  ],
  "metadata": {