I was reading about how the density function $f(x)$ is the "probability per unit length" but was confused what that meant. For context, here is the passage in which I encountered that phrase:
To connect density $f(x)$ with probabilities, we need to look at a very small interval $[x, x+d x]$ close to $x$; then we have $$ \mathbb{P}[x \leq X \leq x+d x]=\int_x^{x+d x} f(z) d z \approx f(x) d x $$ Thus, we can interpret $f(x)$ as the "probability per unit length" in the vicinity of $x$
I have 2 questions here:
- What specifically is meant by 'unit' here? I thought that $dx$ was the smallest unit of anything possible in the Leibniz sense, so it cannot be divided into further units.
- Does this say that $f(x)$ is the average probability in each 'unit' of $dx$, or the precise probability in each 'unit'?
Thanks
When we have a continuous interval like $[a,b]=\{x: a\le x\le b\}$, the probability of getting any specific value is zero. This is because we have one outcome out of an uncountable number of outcomes.
Therefore, instead of asking what the probability is that we get a certain value in an interval, we compute the probability of getting any value in a small interval $[x, x+\delta x]$.
The integral $$\int_{x}^{x+\delta x}f(z)dz$$ is the area under the curve $f$ which is a probability distribution.
The integral represents the probability of getting a value in the interval $[x,x+\delta x]$ so that $f$ has units of probability per unit length and $dz$ has units of length. Therefore, we say that $f$ is a probability density function rather than a probability function.
Note that when we say "per unit length", we just mean probability per whatever unit we use for length. This could be feet, meters or any other unit of length. It is also an instantaneous rate of change because as $\delta x$ gets smaller we get the limiting case.
If we fix $\delta x$, then we get an average. Compare this with computing the slope of a tangent line. We take a secant line and let $\Delta x$ get smaller. It is the limit that gives us the instantaneous change which is why we say $f(z)$ is the probability per unit length at $z$.