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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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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