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Assume that the distribution of the radius $R$ of stars has a density function $f_R$. Find formulas for the density and the distribution function of their volume $V = (4/3)\pi R^3$.

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closed as too localized by t.b., cardinal, Benjamin Lim, Did, J. M. Apr 27 '12 at 12:30

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Will you please try these problems before posting them here? For instance, did you cover the change of variable formula for densities? Please work this problem out from the tools you have--tell us where you're stuck; what are tools you did try to apply and failed and such things. –  user21436 Apr 18 '12 at 5:09
This does not look too localized to close, but, this need not be answered in present form IMO. –  user21436 Apr 18 '12 at 5:11
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Related meta thread: meta.math.stackexchange.com/questions/4001/… –  cardinal Apr 19 '12 at 23:56

1 Answer 1

up vote 1 down vote accepted

Let $F_V$ be the cumulative distribution function of the volume. We have $$F_V(v)=P(V \le v)=P(4\pi R^3/3)\le v)=P(R\le (3v/4\pi)^{1/3}).$$ To save space, let $g(v)=(3/4\pi)^{1/3}v^{1/3}$. For $v\le 0$, we have $F_V(v)=0$. For $v>0$, $$F_V(v)=\int_0^{g(v)} f_R(r)\,dr.$$ Differentiate. By the usual method for differentiating under the integral sign, if $v>0$ then $$f_V(v)=g'(v)f_R(g(v)).$$ It is easy to calculate $g'(v)$. One cannot do much more, since $f_R$ has not been given explicitly.

Note that the procedure we used is quite general. If $X$ is a random variable with known density function, and $h(x)$ is an increasing function, we can use a very similar procedure to find the density function of $h(X)$. For an explicit formula, we need an explicit expression for the inverse function $h^{-1}(x)$ of $h(x)$.

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