mathematics//statistics//normal distribution//central limit theorem
Why sums of many small causes end up shaped like a bell. The noise of a sensor almost never has one cause. It is the sum of many small, independent disturbances: the thermal agitation of electrons, shot noise, fluctuations of the power supply, the last bit of the converter. And the sum of many small independent things tends to a normal almost regardless of what each one looks like:
Why sums of many small causes end up shaped like a bell. The noise of a sensor almost never has one cause. It is the sum of many small, independent disturbances: the thermal agitation of electrons, shot noise, fluctuations of the power supply, the last bit of the converter. And the sum of many small independent things tends to a normal almost regardless of what each one looks like:
v=u1+u2+⋯+un⟹v−nμσn →n→∞ N(0,1).v=u_1+u_2+\cdots+u_n\quad\Longrightarrow\quad \frac{v-n\mu}{\sigma\sqrt n}\ \xrightarrow[n\to\infty]{}\ \mathcal N(0,1).v=u1+u2+⋯+un⟹σnv−nμ n→∞ N(0,1).
Each uiu_iui has mean μ\muμ and standard deviation σ\sigmaσ; standardized, their sum converges to the unit bell.
The theorem has two conditions: the uiu_iui are independent, and their variance is finite. Write them down, because both break in the real world. When one cause dominates the rest (a reflected GPS signal in a narrow street), the sum is no longer a sum of many comparable things and inherits the shape of the dominant one. When the causes share a common driver (a temperature that moves all of them), they are not independent. And when a cause has no finite variance, the Cauchy case, no amount of summing produces a bell (Gaussian assumption).