11 Lecture 10, January 29, 2024

We begin Chapter 5 in this lecture!

11.0.1 Chapter 5. Discrete Random Variable

Definition 11.1 (Random Variable) A random variable is a function that maps assigns a real number \(\mathbb{R}\) to each point in a sample space \(S\). That is, \(X\) is a random variable if \[X:S\to \mathbb{R}\].

Definition 11.2 (Range) The values that a random variables takes is called the range of the random variable. We often denote the range of a random variable \(X\) by \(X(S)\).

Definition 11.3 (Discrete random variable) The discrete random variables take integer values, or more generally, values in a countable set (i.e. finite or countably infinite set). That is, its range is a discrete/countable subset of \(\mathbb{R}\).

Definition 11.4 (Continuous random variable) A random variable is continuous if its range is an interval that is a subset of \({\mathbb R}\) (e.g. $[0,1], (0,), {R} $).

Definition 11.5 (Probability (mass) function) The probability (mass) function of a discrete random variable \(X\) is the function \[ f_X(x) = P(X=x),\quad \text{ for } x\in \mathbb{R}, \] which is non-zero at at most countably many values.

Notation: We write \(P(X=x)\) as the shorthanded notation for \(P(\{\omega \in S : X(\omega)=x\})\).

Notation: We can write \(f_X(x)=P(X^{-1}(x))=(P\circ X^{-1})(x)\). We call this as push-forward probability measure.

Note: The definition \(f_X\) is valid for all \(x\), but the value is zero when \(x\) is outside the range of the random variable \(X\). (This is called the null set).

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Properties of probability mass function \(f\):

1. \(f_X(x)\in[0,1]\) for all \(x\), and

2. \(\sum_{x\in X(\omega)}f_X(x) =1\). i.e. sum of the probability on ALL the events equal to \(1\).