I am working on this question:
If we think of the 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). 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% of the time.
(a) Find a formula for the density function of this distribution. Sketch the density function and the cumulative distribution function.
(b) Find the median distance and the mean distance. Near what value of r is an electron most likely to be found?
Is the density function the derivative of the cumulative distribution function?
P'(r)=4r^{2}e^{-2r}
To find the mean distance I believe I use the formula:
\mu =\int_{-\infty }^{\infty}rP'(r)\cdot dr
For the median I am looking for the number $m$ such that:
\int_{m }^{\infty}P'(r)\cdot dr=\frac{1}{2}
I am thinking to find where the value of r that an electron is most likely to be found involves the max of P'(r).