Chapter 2
Random Variables
 
Lectures 4 - 7
In many situations, one is interested in only some aspects of the random experiment. For example, consider the experiment of tossing $ 3$ unbiased coins and we are only interested in number of 'Heads' turned up. The probability space corresponding to the experiment of tossing $ 3$ coins is given by
  $\displaystyle \Omega \ = \ \{(a_1, a_2, a_3) \, \vert \, a_i \in \{H, \, T \} \} \, , {\mathcal F} \ = \ {\mathcal P}(\Omega) $  
and $ P$ is described by
  $\displaystyle P (\{(a_1, a_2, a_3) \}) \ = \ \frac{1}{9} \, . $  
Our interest is in noting $ a_1+a_2+a_3$, i.e., in the map $ (a_1, a_2, a_3) \to a_1 +a_2 +a_3$. So our interest is only on certain function of the sample space. But all functions defined on the sample space are not useful, in the sense that we may not be able to assign probabilities to all basic events associated with the function. So one need to restrict to certain class of functions of the sample space. This motivates us to define random variables.
Definition 2.1 Let $ (\Omega, \ {\mathcal F}, \ P)$ be a probability space. A function $ X : \Omega \ \to \mathbb{R}$ is said to be a random variable if
  $\displaystyle \{ \omega \in \Omega \, \vert \, X(\omega) \leq x \} \in {\mathcal F} \ { \rm for\ all } \ x \in \mathbb{R} \, . $  
Now on, we denote $ \{ \omega \in \Omega \, \vert \, X(\omega) \leq x \}$ by $ \{ X \leq x\}$.