Continuous distributions

What is it?

Formulas

$$\mathcal U(a,b): \tfrac{1}{b-a}, \qquad \operatorname{Exp}(\lambda): \lambda e^{-\lambda x}, \qquad \mathcal N(\mu,\sigma^2): \tfrac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$$

$$\frac{\bar X_N - \mu}{\sigma/\sqrt N} \xrightarrow{\ d\ } \mathcal N(0,1)$$

central limit theorem

Where it shows up in computing

• Queueing theory and performance★★★★★fundamentalOptimization and systems

  Poisson arrivals have exponential inter-arrival times — the basis of M/M/1 models of servers.

• Kalman filter★★★★★fundamentalRobotics and control

  Gaussian noise assumptions make the Kalman filter optimal and closed-form.

Where it shows up in AI

• Generative models★★★★★frequentAI and machine learning

  Diffusion models add Gaussian noise; VAEs use Gaussian latent variables.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them: