Probability Mass

probability-theory

Definition

Probability Mass

Let $P_X$ be the probability distribution of a random variable $X$ with state space $\mathcal{X}$. The probability mass carried by a value $x\in \mathcal{X}$ is the probability of its singleton:

$$m_X(x)=P_X(\{x\})=P(X=x).$$

For a discrete distribution, each possible value is a point in $\mathcal{X}$, and its mass is represented by a vertical lift. The height over $x$ is the probability of the singleton $\{x\}$:

Contrast with Probability Density

Probability mass at $x$ is an actual probability and may be positive:

$$P(X=x)=m_X(x)>0.$$

For a distribution described by a probability density, the density may be positive at $x$ while the singleton still has zero probability, $P(X=x)=0$. Probability is then carried by regions through integration rather than by individual points.

Example

Biased coin

Suppose $X\in \{H,T\}$ records a coin toss with

$$P(X=H)=0.7,P(X=T)=\mathrm{0.3.}$$

The values $H$ and $T$ carry masses $0.7$ and $0.3$. Their total mass is

$$m_X(H)+m_X(T)=1.$$