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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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up vote 1 down vote accepted

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) $P(r)=\frac{1}{2}.$

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Likewise, the mean distance is also the integral of $1-P(r)$ on $r\geqslant0$. –  Did Feb 9 '12 at 7:26
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