| [[Discrete random variables]]. | [[Continuous random variables]]. | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------ | -------------------------------------------------------------------------------------------------------- | | $\exists S\subset \mathbb{R}: \|S\|\leq \|\mathbb{N}\|$ such that $\mathbb{P}(X\in S)=1.$ | $\forall S \subset \mathbb{R}:\|S\|\leq \|\mathbb{N}\|\implies \mathbb{P}(X\in S)<1.$ | | [[Probability Distribution of Real-Valued Random Variable\|Distribution]] is fully characterised by [[Probability Mass Function]] $p_{X}:\mathbb{R}\to \mathbb{R}$ | Distribution is fully characterised by [[Probability Density Function]] $f_{X}:\mathbb{R}\to \mathbb{R}$ | | $\forall x\in \mathbb{R}:p_{X}(x)\geq 0$ | $\forall x\in \mathbb{R}:f_{X}(x)\geq 0$ | | $\mathbb{P}(X=x)=p_{X}(x)$ | $\mathbb{P}(X=x)=0$ | | $\mathbb{P}(a\leq X\leq b)= \sum_{a\leq x\leq b}p_{X}(x)$ | $\mathbb{P}(a\leq X\leq b)= \int _{a}^{b} f_{X}(x) \, dx$ | | $p_{X}(x)=\mathbb{P}(X=x)$ | defined implicitly by above | | $\sum_{x}p_{X}(x)=1$ | $\int_{-\infty }^{\infty}f_{X}(x)\,dx=1$ | | $\forall x\in \mathbb{R}:p_{X}(x)\leq 1$ | $f_{X}(x)$ may be gt;1$ | | $\mathbb{E}[g(X)]=\sum_{x}g(x)p_{X}(x)$ | $\mathbb{E}[X]=\int_{-\infty}^{\infty}g(x)f_{X}(x) \, dx$ |