If we think of an electron as a particle, the function $P(r)=1-(2r^2 + 2r +1)e^{-2r}$ is the cumulative distribution function of the distance, $r$, of the electron in a hydrogen atom from the center of the atom. The distance is measured in Bohr radii. (1 Bohr radius $= 5.29 10^{-11}$ meters. Niels Bohr (1885-1962) was a Danish physicist.)

For example, $P(1) = 1-5e^{-2} \approx 0.32$ means that the electron is within $1$ Bohr radius from the center of the atom $32$ per cent of the time.

1. Find a formula for the density function of this distribution. Sketch the density function and the cumulative distribution function.
   $p(r) =$
   
   
2. Find the median distance and the mean distance. Near what value of $r$ is an electron most likely to be found?
   Mean =
   Median =
   Most likely distance =

You should be able to explain why the Bohr radius is sometimes called the "radius of the hydrogen atom".