Density curves and area as probability

From a list to a curve

Module 6's random variables took separated values — 0, 1, 2, 3 late invoices — and a table of probabilities described them. A continuous random variable takes any value in an interval: the minutes a client waits on the phone, the dollars on an invoice, the days until a cheque clears. There is no table, because there are infinitely many values to list. Instead a density curve $f(x)$ describes the variable, and probability is area under the curve:

$$P(a<X<b)=\text{area under }f(x)\text{ between }a\text{ and }b$$

Two facts make a curve a density: $f(x)\ge 0$ everywhere, and the total area under the curve is 1 — the continuous version of Module 6's "the probabilities sum to one".

Worked example — a uniform wait

Northfield Credit promises that a call is answered within 20 minutes and, from its call logs, the wait $X$ is equally likely to be anywhere in that window: $X$ is uniform on $[0,20]$. The density is flat — a rectangle of width 20 whose height must make the area 1, so the height is $1/20=0.05$.

The probability a caller waits more than 15 minutes is the area of the strip from 15 to 20: width 5, height 0.05,

$$P(X>15)=5\times 0.05=\frac{5}{20}=0.25$$

The probability of waiting between 5 and 12 minutes is $7/20=0.35$. Any interval's probability is its width as a fraction of 20.

A line has no width

What is the probability the wait is exactly 10 minutes — 10.000…, not 10.01? A single value is a line under the curve, and a line has zero width, so zero area:

$$P(X=10)=0$$

This is the one place continuous variables behave unlike discrete ones. It does not mean a 10-minute wait is impossible; it means the probability mass is spread over an interval too thinly for any one point to carry a positive share. The consequence the paper tests: for a continuous $X$,

$$P(X<10)=P(X\le 10),P(X>15)=P(X\ge 15)$$

because the two sides differ only by $P(X=c)=0$. "At most" and "fewer than" — Module 6's whole boundary discipline — collapse to the same number. Keep the discipline for the discrete case; drop it, deliberately, for the continuous one.

The cumulative distribution function

The cdf is the running area from the far left up to $x$:

$$F(x)=P(X\le x)$$

For the uniform wait, $F(15)=15/20=0.75$, and $P(X>15)=1-F(15)=0.25$ — the complement rule on areas. An interval is a difference of two cdf values: $P(5<X<12)=F(12)-F(5)=0.60-0.25=0.35$. Every normal probability in L7.3 is computed exactly this way; the only change is that the cdf comes from a table instead of a fraction.

The uniform is the one continuous distribution simple enough to do by hand. It and the exponential appear in IBS1 as examples; this course uses the uniform only to fix the idea of area, and moves to the normal — the distribution every later module needs — in L7.2.