# Integral and Density/Cumulative Distribution Function

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 $P'(r)$.

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What you have written is sensible, but since you have the cumulative distribution function you can find the median directly by solving (numerically or looking at your graph) $P(r)=\frac{1}{2}.$
Likewise, the mean distance is also the integral of $1-P(r)$ on $r\geqslant0$. – Did Feb 9 '12 at 7:26