Normal Distribution & Probability Density
Q: What does Density functions compute?
The Normal (Gaussian) PDF: $f(x) = \frac{1}{\sigma\sqrt{2\pi}} \cdot e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$
Returns the relative likelihood of a value given a distribution defined by mean (μ) and std (σ).
Q: If I integrate from -∞ to +∞, what happens?
The result is exactly 1 — total probability over all outcomes = 100%. The $\frac{1}{\sigma\sqrt{2\pi}}$ constant exists precisely to ensure this.
Q: Shouldn't the density value at each point be 0?
That's the probability of an exact point, yes — but density ≠ probability. Density is a relative score (like kg/m³). The probability of a range is obtained by integrating density over that range.
Q: So integration of density = probability?
Yes: $P(a \leq X \leq b) = \int_a^b f(x)dx$
| Need | Approach |
|---|---|
| Is this value anomalous? | Compare density scores directly |
| Probability of falling in a range | Integrate density over that range |
| Rank anomalous tenants | Sort by density — lowest = most anomalous |
Q: Why express probabilities via density at all?
For continuous variables, assigning non-zero probability to every point is impossible — uncountably infinite points would sum to ∞. Density sidesteps this by answering "how likely is x relative to other values?" rather than assigning absolute probability to a point.