The theoretical distribution function

An empirical mean and histogram summarize one finite list of trials. Measure height again tomorrow and the table moves. What object stays with the random variable $X$ itself — a rule that, for any fixed number $x$, says how likely $X$ is to fall below $x$?

Probability to the left of $x$

That rule is the theoretical distribution function $\Phi$ of $X$:

$$\Phi (x):=P(X<x).$$

So $\Phi (x)$ is the probability that the random variable is strictly less than a given number $x$. One sample’s $\bar{x}$ and $\Delta x$ estimate location and spread; $\Phi$ is the full left-tail law those summaries are aiming at.

The bell peaks at $\mu$

Many measured quantities follow a normal (Gaussian) law. Its probability density is the bell

$$\phi (x):=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-\mu {)}^2/(2{\sigma }^2)}.$$

The curve reaches its maximum at $x=\mu$. The positive number $\sigma$ is the standard deviation: the smaller $\sigma$ is, the more the mass concentrates at $\mu$. Here $\mu$ and $\sigma$ are the theoretical mean and standard deviation of $X$ — the targets that the empirical $\bar{x}$ and $\Delta x$ estimate from a finite list.

Drag $s$. At larger $s$ the blue bell is wide; drag down and the same total mass piles nearer the blue mark at $\mu =0$. The green marks track $\mu \pm \sigma$. That is the concentration claim on the board: shrink $\sigma$ and the density sharpens at the mean.

Area is probability

Under a normal density, probability is area. The shaded area between vertical lines at $a$ and $b$ equals the probability that $X$ lies in the interval $[a,b]$. In terms of the distribution function that area is exactly

$$\Phi (b)-\Phi (a).$$

The single value $\Phi (a)$ is the area under the bell to the left of $a$.

Drag $a$ and $b$ on the standard normal ($\mu =0$, $\sigma =1$). The orange shade is the living $\Phi (b)-\Phi (a)$. Widen the gap and the probability grows; slide both marks into a thin window and the shade collapses.

The same density also has an S-shaped cumulative picture: plot $\Phi (x)$ against $x$ and you get a smooth step from $0$ toward $1$, with $\Phi (a)$ read as the height at $a$.

Drag $a$. The green curve is $\Phi$ for the standard normal; the orange height is $\Phi (a)$. That height is the left-tail area from the previous board, redrawn as a single number on the vertical axis.

A window of probability $1-\alpha$

Fix a small $\alpha >0$. An $\alpha$-confidence interval $[x_{\alpha }^{-},x_{\alpha }^{+}]$ for $X$ is a window that traps the random variable with probability $1-\alpha$:

$$x_{\alpha }^{-}\le x\le x_{\alpha }^{+}$$

holds with probability $1-\alpha$. For a normal law the natural window is symmetric about $\mu$. Its endpoints are

$$x_{\alpha }^{+}=\mu +\sigma z_{\alpha },x_{\alpha }^{-}=\mu -\sigma z_{\alpha },$$

where the factor $z_{\alpha }$ comes from a short table:

So the random variable $X$ lies in the $\alpha$-confidence interval with probability $1-\alpha$.

Drag $z$ with $\mu =10$ and $\sigma =2$. At $z=2.6$ ($\alpha =0.01$) the orange shade runs from $10-2\cdot 2.6=4.8$ to $10+2\cdot 2.6=15.2$. At $z=2.0$ or $z=1.6$ the window shrinks to the $5\%$ and $10\%$ rows of the table. In a long run of trials, about $(1-\alpha )n$ of the measured values fall inside that window — for $n=1000$ and $\alpha =0.01$, about $990$ of them between $4.8$ and $15.2$.

What this page put in place

Beyond one empirical list, the theoretical distribution $\Phi (x)=P(X<x)$ is the left-tail law of $X$. For a normal random variable the density is the Gaussian bell with mean $\mu$ and standard deviation $\sigma$; areas under that bell are probabilities, including $\Phi (b)-\Phi (a)$ on an interval. An $\alpha$-confidence interval $\mu \pm \sigma z_{\alpha }$ then names a window of probability $1-\alpha$, with $z_{\alpha }$ read from the table above. The next natural question is how to check whether a given measurement sequence is compatible with such a normal law — that step is checking for a normal distribution.