Dr. Brajesh Kumar Jha

Discrete and Continuous Probability Functions 1. Discrete Probability Function Let \(X\) be a discrete random variable. A function \[ p(x)=P(X=x) \] is called the probability mass function (PMF) or discrete probability function of \(X\). For a function \(p(x)\) to be a valid discrete probability function, it must satisfy the following conditions: \[ \boxed{p(x)\geq 0} \] and \[ \boxed{\sum_x p(x)=1}. \] Thus, the probability of an event such as \(a\leq X\leq b\) is obtained by adding the corresponding probabilities: \[ P(a\leq X\leq b) = \sum_{x=a}^{b}p(x). \] 5 Solved Examples on Discrete Probability Functions Example 1: Determine the Constant in a PMF A discrete random variable \(X\) has probability function \[ P(X=x)=kx,\qquad x=1,2,3,4. \] Find: The value of \(k\) \(P(X\leq 2)\) The mean \(E(X)\) Show Solution Since the total probability must be equal to 1, \[ \sum_{x=1}^{4} P(X=x)=1. \] Therefore, \[ k...

Random Variable: Definition, Types and 10 Solved Examples Definition of Random Variable A random variable is a real-valued function defined on the sample space of a random experiment. \( X:S\rightarrow \mathbb{R} \) It assigns a numerical value to every possible outcome of a random experiment. Example Suppose two coins are tossed. The sample space is \( S=\{HH,HT,TH,TT\}. \) Let \(X\) denote the number of heads obtained. Then \(X(HH)=2,\quad X(HT)=1,\quad X(TH)=1,\quad X(TT)=0.\) Therefore, \(X\in\{0,1,2\}.\) Types of Random Variables 1. Discrete Random Variable A random variable is called discrete if it takes a finite or countably infinite number of values. Its probability mass function (PMF) is \(p_X(x)=P(X=x)\) and it satisfies \(p_X(x)\geq 0,\qquad \sum_x p_X(x)=1.\) 2. Continuous Random ...