Let (X, Y) be uniformly distributed in the circle of radius...

Let $(X, Y)$ be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus $$ f(x, y)=\frac{1}{\pi} \quad 0 \leq x^{2}+y^{2} \leq 1 $$ Let $R=\left(X^{2}+Y^{2}\right)^{1 / 2}$ and $\theta=\tan ^{-1}(Y / X)$ denote its polar coordinates. Show that $R$ and $\theta$ are independent with $R^{2}$ being uniform on $(0,1)$ and $\theta$ being uniform on $(0,2 \pi)$.

Let $(X, Y)$ be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus
$$
f(x, y)=\frac{1}{\pi} \quad 0 \leq x^{2}+y^{2} \leq 1
$$
Let $R=\left(X^{2}+Y^{2}\right)^{1 / 2}$ and $\theta=\tan ^{-1}(Y / X)$ denote its polar coordinates. Show that $R$ and $\theta$ are independent with $R^{2}$ being uniform on $(0,1)$ and $\theta$ being uniform on $(0,2 \pi)$.

Key Concepts

Independence of Random Variables

Random variables are independent when the joint density function factors into a product of the marginal densities. Demonstrating independence often involves showing that after an appropriate transformation, the resulting variables do not interact in the joint density, meaning that knowing one provides no information about the other.

Jacobian Determinant

The Jacobian determinant is a crucial factor in the change-of-variables formula in multivariate integrals. It accounts for the scaling and distortion introduced by the transformation, ensuring that probabilities remain consistent when switching between coordinate systems.

Distribution of Functions of Random Variables

Determining the distribution of a function of a random variable involves transforming the original variable and calculating the corresponding density or cumulative distribution function. This process is essential when re-expressing a measure in a more convenient form, such as showing that a transformed radius squared follows a uniform distribution.

Uniform Distribution

Uniform distributions assign an equal probability density to all outcomes in a given set. In multidimensional settings, such as areas or volumes, the density is constant over the specified region, which simplifies the calculation of probabilities and marginal distributions, especially when the region has a symmetry that can be exploited using appropriate coordinate transformations.

Transformation of Variables (Change to Polar Coordinates)

Changing variables from Cartesian to polar coordinates is a common technique when dealing with radially symmetric regions. This transformation simplifies the analysis by aligning the coordinate system with the natural symmetry of the problem, allowing the separation of variables into independent components such as radius and angle.

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