Example formalization
Below is the probability mass function of an unfair die, where we observe a 1, 2, or 3 with probability \(\frac{1}{12}\), and a 4, 5, or 6 with probability \(\frac{1}{4}\). The PMF can therefore be defined as:
\[\begin{equation} f(x) = \begin{cases} \frac{1}{12} & : x = 1 \\ \frac{1}{12} & : x = 2 \\ \frac{1}{12} & : x = 3 \\ \frac{1}{4} & : x = 4 \\ \frac{1}{4} & : x = 5 \\ \frac{1}{4} & : x = 6 \\ 0 & : otherwise \\ \end{cases} \end{equation}\]What is \(Pr[X \geq 2]\)?
\[Pr[X \geq 2] = \] \[\sum_{x = 2}^6 f(x) = \] \[\frac{1}{12} + \frac{1}{12} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \] \[\frac{11}{12}\]
Example visualization
Example visualizations
\[ \Phi(x) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}}e^\frac{-u^2}{2}du \]
Example visualization