Added PMF notebook

notebooks/probability/pmf.ipynb

@@ -2,7 +2,7 @@
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     "### Probability Mass Function (PMF)\n",
@@ -17,48 +17,46 @@
     "\n",
     "We can denote the sample space as: $\\Omega = \\{H, T\\}$\n",
     "\n",
-    "Now, let us say that if the coin lands as Heads, we win 1\\$ and we win 0\\$ if the coin lands as Tails. We can associate a function $W$ with the experiment such that W(Heads) = 1 and W(Tails) = 0\n",
+    "Now, let us consider a discrete random variable $X$ which takes the values 1 and 0 if the coin lands as Heads and Tails respectively, i.e, $X$(Heads)  =  1 and $X$(Tails)  = 0. Thus, $X$ can be used to represent the occurence of Heads in a single toss of a coin. \n",
     "\n",
+    "Let, $R_{X}$ be the range of X, then \n",
+    "    <center>$R_{X} = \\{0, 1\\}$.\n",
     "\n",
-    "```{admonition} Random Variable\n",
-    "A random variable is a function mapping the sample space (for example: $\\Omega = \\{H, T\\}$) of an experiment to a measurement space $E = \\{0, 1\\}$. Random variables are represented using capital letters.\n",
-    "```\n",
-    "\n",
-    "While we may have generally seen a random variable W associated with a coin toss taking the values 1 for Heads, 0 for Tails, we might have chosen any other values and still have a random variable. As an example\n",
-    "\n",
-    "$$\n",
-    "Y(\\omega)= \\begin{cases}100, & \\text { if } \\omega=\\text { heads } \\\\ -20, & \\text { if } \\omega=\\text { tails }\\end{cases}\n",
-    "$$\n",
-    "\n",
-    "\n",
-    "If our experiment is to roll two die (6-faced die) and note the sum of the numbers on the top side.\n",
+    "Then, Probability Mass Function(PMF) of discrete random variable $X$ gives the probability distribution for each vaule of $X$.\n",
+    "    \n",
+    "<center>$P_{X}(0) = P_{X}(X=0) = \\frac{1}{2}$ when Heads occurs <br>\n",
+    "        $P_{X}(1) = P_{X}(X=1) = \\frac{1}{2}$ when Heads does not occur\n",
     "\n",
-    "The sample space for this example is: \n",
-    "\n",
-    "$$\n",
-    "\\begin{align*}\n",
-    "    S = \\{&(1, 1), (1, 2), \\dots, (1, 6), \\\\\n",
-    "          &(2, 1), (2, 2), \\dots, (2, 6), \\\\\n",
-    "          &\\cdots, \\cdots,  \\cdots, \\cdots \\\\\n",
-    "          &(6, 1), (6, 2), \\dots, (6, 6) \\}\n",
-    "\\end{align*}\n",
-    "$$\n",
-    "\n",
-    "Based on this sample space, we can see that our random variable $Z$ denoting the sum of numbers on the top side takes the values $\\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\\}$\n",
+    "```{admonition} Probability Mass Function (PMF)\n",
+    "Probability mass function also called the probability law is the probability distribution of a discrete random variable, and provides the possible values and their associated probabilities.It is the function $P : \\mathbb{R} → [ 0 , 1 ]$  defined by <br>\n",
+    "$P_{X}(x_k) = P_{X}(X=x_k)$ where $k = 1,2,3,\\dots$<br>\n",
+    "where $X$ is a discrete random variable with range $R_{X} = \\{x_1,x_2,x_3,\\dots\\}$ (finite or countably infinite).\n",
+    "```\n",
+    "<br>\n",
+    "Example. Let us consider tossing a fair coin twice and finding the probability mass function (PMF) of a random discrete variable $Y$ which observe the number of occurences of Heads in each event.\n",
     "\n",
+    "The sample space for this example is:\n",
+    "    <center>$S=\\{HH,HT,TH,TT\\}$.\n",
     "\n",
-    "```{admonition} Discrete Random Variable\n",
-    "A discrete random variable can take discrete values. For example, W representing the money we win if we toss a coin randomly. Or, Y the top face of a dice when the dice is rolled. Or, $Z$ denoting the sum of numbers on the top side of two dies.\n",
-    "```\n",
+    "The number of Heads will be 0, 1 or 2. Thus \n",
+    "    <center>$R_{Y} = \\{0, 1, 2\\}$.    \n",
+    "    \n",
+    "Since, it is a finite set, thus the variable $Y$ is discrete random variable. Therefore, the PMF of $Y$, $P_Y$ is defined as \n",
     "\n",
+    "<center>$P_{Y}(k) = P_{Y}(Y=k)$ for $k = 0,1,2$.\n",
     "\n",
-    "Let us now take a different example. Our experiment is to pick a person at random from a university and measure their weight. The sample space would be the list of all the people in the university and the random variable $T$ would be weight corresponding to that randomly chosen person. The weight of that person would be a continuous scale.\n"
+    "We have,\n",
+    "<center>$P_{Y}(0) = P_{Y}(Y=0) = P(TT) = \\frac{1}{4}$,<br>\n",
+    "$P_{Y}(1) = P_{Y}(Y=1) = P(\\{HT,TH\\}) = \\frac{1}{4} + \\frac{1}{4} = \\frac{1}{2}$,<br>\n",
+    "$P_{Y}(2) = P_{Y}(Y=2) = P(HH) = \\frac{1}{4}$.\n",
+    "    \n",
+    "    "
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@@ -72,7 +70,7 @@
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