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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} $

  • 1
    Since Shai Covo asked if this was homework (without getting a reply),$I$will point out that this is an end-of-chapter problem in Sheldon Ross's _A First Course in Probability_ (Problem 6.29 in 6th edition).2012-04-01

3 Answers 3

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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]}. $

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    @StephenDedalus One direction is obvious. In the other direction, note that the functions $h_t:w\mapsto\exp(itw)$ are bounded continuous hence if the condition holds for every such function it holds for every $h_t$ hence one knows the Fourier transform of the distribution of $W$, QED.2014-09-01
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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.

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    In the past, I was successful in solving this variety of delta function integrals in Mathematica using one of the 'limit definitions' of the delta function. In my case, computing the integral using the Lorentzian limiting function and then taking this result and applying the delta function limit to the Lorentzian parameters.2017-10-30
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$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}}$ $\ds{\Theta\pars{x}}$ 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}} $

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