Point Mass¶

Definition¶

A distribution has a point mass, or atom, at $a\in\mathbb{R}$ when $p_a=\mathbb{P}(X=a)>0$$0$. Here $X$ is a real-valued random variable with that distribution. The CDF has a jump of size $p_a$ at $a$:

Interpretation¶

A point mass is a single value that carries positive probability on its own. On the CDF it shows up as a jump: just to the left of $a$ the CDF approaches $F_X(a^-)$, at $a$ it equals $F_X(a)$, and the height of the step is exactly $\mathbb{P}(X=a)$$a)$.

Because $F_X$ is right-continuous, the mass at $a$ is included in $F_X(a)=\mathbb{P}(X\le a)$$a)$, while the left limit $F_X(a^-)$ equals $\mathbb{P}(X<a)$$a)$. With atoms present, a strict and a non-strict inequality can therefore give different probabilities.

Every value with positive PMF of a discrete random variable is a point mass, and these masses add up to one. A point mass can also be only one part of a distribution: in the Example, $0.3$ sits at zero and the remaining $0.7$ is spread by a density, a mixture of a discrete and a continuous part. An atom at $a$ therefore does not mean that all probability sits at $a$.

A continuous random variable has a CDF without jumps, so $\mathbb{P}(X=a)=0$ for every $a$, even where the density is large: density height is not point probability.

Examples¶

Suppose $X=0$ with probability $0.3$ and, with probability $0.7$, $X$ is drawn from the uniform distribution on $(0,1)$. Then $F_X(0^-)=0$ and $F_X(0)=0.3$$0.3$: there is a point mass $0.3$ at zero. The remaining probability is spread over the interval.

This differs from a constant random variable, for which the point mass at its only possible value is one. An absolutely continuous distribution has no point masses.

Prerequisites¶

• Distribution
• Cumulative Distribution Function

Related Concepts¶

• Discrete Random Variable
• Probability Mass Function
• Absolutely Continuous Distribution

Sources¶

• Wiki terminology supplement to the PT CDF convention; placement alongside Week 5 is provisional pending course review.
• Marco Taboga, Distribution function, discrete CDFs and cumulative probability masses.

Aliases: atom of a distribution, atom of a probability distribution, probability atom Symbols: element of, real numbers

Last updated: 22 October 2026 19:04