How to deduce the CDF of $W=I^2R$ from the PDFs of $I$ and $R$ independent

Question

Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find cdf of $W =I^2R$?

Where,

$$ \begin{align} f_I(i)&=6i(1-i), &0 \leq i \leq 1 \\ f_R(r)&=2r, &0 \leq r\leq 1. \end{align} $$

Answer 1

The simplest and surest way to compute the distribution density or probability of a random variable is often to compute the means of functions of this random variable. In the case at hand, one wants to write $\mathrm E(g(W))$ as $$ \color{blue}{\mathrm E(g(W))=\int g(w)f(w)\mathrm{d}w}, $$ for every bounded measurable function $g$, then one can be sure that $f$ is the density of the distribution of $W$. So, in a way, the functions $g$ play the role of a dummy variable and one wants the equality above to hold for every $g$.

Naturally $W=I^2R$ hence $\mathrm E(g(W))$ is a priori a double integral, but one can be sure that a change of variable will save the day. So, applying the definitions, $\mathrm E(g(W))=\mathrm E(g(I^2R))$ and $$ \mathrm E(g(I^2R))=\iint g(x^2y)\cdot[0\leqslant x\leqslant 1]\cdot6x(1-x)\cdot[0\leqslant y\leqslant 1]\cdot2y\cdot\mathrm{d}x\mathrm{d}y, $$ where, for every property $\mathfrak{A}$, Iverson bracket $[\mathfrak{A}]$ denotes $1$ if $\mathfrak{A}$ holds and $0$ otherwise.

(Begin of rant: no, I do not like to put the limits of the domain of integration on the integral signs, and yes, I prefer to use the notation $[\mathfrak{A}]$ or its cousin $\mathbb{1}_\mathfrak{A}$ because they are more systematic and, at least to me, less error prone. End of rant.)

Now, what change of variable? For one of the two new variables, we want $w=x^2y$, of course. For the other, a sensible choice (but not the only one) is $z=x$. The new domain is $0\leqslant w\leqslant z^2\leqslant 1$ and the Jacobian is given by $\mathrm{d}x\mathrm{d}y=z^{-2}\mathrm{d}w\mathrm{d}z$, hence $$ \mathrm E(g(W))=\int g(w)[0\leqslant w\leqslant 1]\left(\int [w\leqslant z^2\leqslant 1]\cdot6z(1-z)(2wz^{-2})z^{-2}\mathrm{d}z\right)\mathrm{d}w. $$ By identification, the density $f(w)$ is the quantity enclosed by the parenthesis, that is, for every $0\leqslant w\leqslant1$, $$ f(w)=\int [w\leqslant z^2\leqslant 1]6z(1-z)(2wz^{-2})z^{-2}\mathrm{d}z=12w\int_{\sqrt{w}}^1 z^{-3}(1-z)\mathrm{d}z, $$ Finally, $$ \color{red}{f(w)=6(1-\sqrt{w})^2\cdot[0\leqslant w\leqslant1]}. $$

Answer 2

Probability w = W is probability I^2 R = W.

$$f_W(w) = \int \delta(w - i^2 r) f_{I,R}(i, r) \, di \, dr$$

Independence means that $f_{I,R}(i, r) = f_I(i) f_R(r)$.

(I suggest doing the R integral first -- the delta function transformation is easier.)

Changing to the cumulative distribution function is just integration.

$$F_W(w_0) = \int_0^{w_0} f_W(w) dw.$$

Of course, you can plug the first one into the second, and do the W integral first. This is nice as it handles the delta function quite easily.

Answer 3

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{\Theta\pars{x}}$ and $\ds{\delta\pars{x}}$ are the Heaviside Step Function and the Dirac Delta Function, respectively.

\begin{align} {\rm P}\pars{W}&=\totald{}{W}\int_{0}^{W}{\rm P}\pars{t}\,\dd t= \totald{}{W}\int_{0}^{1}6I\pars{1 - I} \int_{0}^{1}2R\,\Theta\pars{W - I^{2}R}\,\dd R\,\dd I \\[3mm]&= 12\int_{0}^{1}I\pars{1 - I}\int_{0}^{1}R\,\delta\pars{W - I^{2}R}\,\dd R\,\dd I \\[3mm]&= 12\int_{0}^{1}I\pars{1 - I}\int_{0}^{1}R\,{\delta\pars{R - W/I^{2}} \over I^{2}} \,\dd R\,\dd I \\[3mm]&=12\int_{0}^{1}{1 - I \over I}\,{W \over I^{2}} \int_{0}^{1}\delta\pars{R - {W \over I^{2}}}\,\dd R\,\dd I \\[3mm]&=12W\int_{0}^{1}{1 - I \over I^{3}}\, \Theta\pars{1 - {W \over I^{2}}}\,\dd I =12W\int_{0}^{1}{1 - I \over I^{3}}\,\Theta\pars{I - \root{W}}\,\dd I \\[3mm]&=12W\,\Theta\pars{1 - W}\int_{\root{W}}^{1}{1 - I \over I^{3}}\,\dd I =12W\,\Theta\pars{1 - W}\,{\pars{1 - \root{W}}^{2} \over 2W} \end{align}

$$\color{#00f}{\large% {\rm P}\pars{W} = \Theta\pars{W}\Theta\pars{1 - W}6\pars{\root{W} - 1}^{2}} $$