Hair Today, Gone Tomorrow
It’s been quite a long time since I last cut my hair. It’s now past my shoulders, where it has remained in length for the last several years.
Even though it’s still growing, it isn’t getting any longer because it also falls out. It has reached a dynamic equilibrium, where the rate of hair growing in equals the rate falling out.
To better understand this process, I looked up a few facts about hair growth. Hair grows in at a rate of ~0.5 inches/month. There are about 100,000 hairs on a human head, of which about 100 fall out every day. This comes out to 3% of hairs falling out per month. There are other nuances to hair, but I am simplifying the model to assume that growth is constant and that all strands have the same chance of falling out.
I simulated this dynamic in R and plotted a graph to see how hair length changes over time. It starts without hair (imagine a buzz cut) and converges to a steady state length of about 16 inches. The y-axis is reversed to make the plot itself look like hanging strands of hair.
Suppose T is a random variable that represents time until a strand of hair falls out, or the age of a hair. In statistics, the hazard rate is the instantaneous rate at which some event, like failure or death, happens given that it has not yet happened.
By applying the definition of conditional probability,
and taking the limit as dt → 0, the hazard rate is the ratio of the probability density function (pdf) to 1- the cumulative density function (cdf). A hazard rate uniquely characterizes a distribution, and vice versa. A constant hazard rate is a unique property of the exponential distribution.
Here, the hazard rate is .03/month, so T ~ Exp(rate = .03/month). Hair ages are then exponentially distributed with a mean age of 1/.03 = 33 months. Since length grows linearly with age, hair length ~ Exp(rate = .06/inch) and has mean length 16.7 inches. This agrees with the mean hair length converged to in the simulation. The Q-Q plot compares hair lengths at the last time step of the simulation with randomly generated exponentials, and confirms they have the same distribution.