1.6 Discrete Random Variables

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Definition 1.6.1: Discrete random variable

Definition

Let

• $X$ : Random variable
• $\mathcal{D}$ : Space of $X$

If $\mathcal{D}$ is either finite or countable

Then we say $X$ is a discrete random variable

Definition 1.6.2: Probability mass function (pmf)

Definition

Let

• $X$ : Discrete random variable
• $\mathcal{D}$ : Space of $X$

If $ p_{X}(x) = P[X=x], \quad \forall x\in \mathcal{D} $$

Then we say that $p_X(x)$ is the probability mass function (pmf) of $X$

Definition: Support of discrete random variable

Let

• $X$ : Discrete random variable
• $\mathcal{D}$ : Space of $X$

If

$${\mathcal{S}}_X=\{x\in \mathcal{D}:P(X=x)>0\}$$

Then we say ${\mathcal{S}}_X$ is the support of $X$