Dirac Distribution

A Dirac (or degenerate) distribution on a Probability Space with support only on a lower-dimensional manifold, which could be a point. Usually, we consider $\Omega \subset \mathbb{R}$ and the support is a point.

• Notation
  • ${\delta }_{x_0}$
• Parameter
  • $x_0\in \mathbb{R}$
• PMF
  • $p(x)=\delta (x-x_0)$ ^pdf
    • $\delta$ is the Dirac delta function, which is 0-valued everywhere except 0, yet integrates to 1.
    • By defining $p(x)$ to be $\delta$ shifted by $-x_0$ we obtain an infinitely narrow and infinitely high peak of probability mass where $x=x_0$.
• CDF
  • $F(x)=1\left\{x\ge x_0\right\}$
• Mean
  • $x_0$
• Variance
  • $0$
• MGF
  • $e^{x_{0}t}$