Let $R$ be a random variable with exponential distribution of parameter $1/2$, and let $\Theta$ be independent of $R$ with uniform distribution on $[0, 2\pi]$. **Question:** What is the law of $X = \sqrt{R}\cos(\Theta)$? Write $X = \sqrt{R}\cos\Theta$ and $Y = \sqrt{R}\sin\Theta$, and use the 2D change-of-variables formula to find the joint density $f_{X,Y}$. You will need the inverse map expressing $(R, \Theta)$ in terms of $(X, Y)$. Notice that $X^2 + Y^2 = R$. Compute the Jacobian determinant of the inverse map and check whether the resulting joint density factors as a function of $x$ times a function of $y$. ### Reminder: 2D change of variables Let $(R,\Theta)$ have joint density $f_{R,\Theta}$, and let $(X,Y) = T(R,\Theta)$ where $T$ is a smooth bijection with smooth inverse $T^{-1}$. Then the joint density of $(X,Y)$ is : $$ \boxed{ f_{X,Y}(x,y) = f_{R,\Theta}\big(T^{-1}(x,y)\big)\,\left| J_{T^{-1}}(x,y) \right| } $$ where $J_{T^{-1}}$ is the Jacobian determinant of the inverse map: $$ J_{T^{-1}}(x,y) = \det\begin{pmatrix} \dfrac{\partial r}{\partial x} & \dfrac{\partial r}{\partial y} \\ \dfrac{\partial \theta}{\partial x} & \dfrac{\partial \theta}{\partial y} \end{pmatrix} $$ Independence of $X$ and $Y$ follows if $f_{X,Y}$ factors as a function of $x$ times a function of $y$. ### Step 1: Joint density of $(R,\Theta)$ Since $R \sim \text{Exp}(1/2)$: $$ f_R(r) = \tfrac{1}{2}e^{-r/2}, \quad r \geq 0 $$ Since $\Theta \sim \text{Unif}[0, 2\pi]$: $$ f_\Theta(\theta) = \tfrac{1}{2\pi}, \quad \theta \in [0, 2\pi] $$ By independence, $$ f_{R,\Theta}(r,\theta) = \frac{1}{4\pi}\,e^{-r/2}, \quad r \geq 0,\ \theta \in [0,2\pi] $$ ### Step 2: Define the transformation Set $$ X = \sqrt{R}\cos\Theta, \qquad Y = \sqrt{R}\sin\Theta $$ Since $X^2 + Y^2 = R(\cos^2\Theta + \sin^2\Theta) = R$, the inverse map is $$ R = X^2 + Y^2, \qquad \Theta = \arctan\!\left(\frac{Y}{X}\right) $$ ### Step 3: Jacobian of the inverse map Differentiating $r = x^2 + y^2$ and $\theta = \arctan(y/x)$: $$ \frac{\partial r}{\partial x} = 2x, \quad \frac{\partial r}{\partial y} = 2y, \quad \frac{\partial \theta}{\partial x} = \frac{-y}{x^2+y^2}, \quad \frac{\partial \theta}{\partial y} = \frac{x}{x^2+y^2} $$ Therefore $$ J_{T^{-1}} = \det\begin{pmatrix} 2x & 2y \\ \dfrac{-y}{x^2+y^2} & \dfrac{x}{x^2+y^2} \end{pmatrix} = \frac{2x^2}{x^2+y^2} + \frac{2y^2}{x^2+y^2} = 2 $$ ### Step 4: Joint density of $(X,Y)$ Substituting $r = x^2 + y^2$ and $|J_{T^{-1}}| = 2$ into the change-of-variables formula: $$ f_{X,Y}(x,y) = f_{R,\Theta}(x^2+y^2,\ \theta)\cdot 2 = \frac{1}{4\pi}\,e^{-(x^2+y^2)/2}\cdot 2 = \frac{1}{2\pi}\,e^{-(x^2+y^2)/2} $$ ### Step 5: Factor and conclude The joint density factors: $$ f_{X,Y}(x,y) = \left(\frac{1}{\sqrt{2\pi}}\,e^{-x^2/2}\right)\left(\frac{1}{\sqrt{2\pi}}\,e^{-y^2/2}\right) = f_X(x)\,f_Y(y) $$ Each factor is the density of a $\mathcal{N}(0,1)$ variable, and the density factors as a product. Hence $X = \sqrt{R}\cos\Theta$ and $Y = \sqrt{R}\sin\Theta$ are independent standard normal random variables. so : $$ \boxed{ X \sim \mathcal{N}(0,1) } $$

Box-Muller Transformation

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