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Magazine Chapter 6: Problem 52
Let \(X\) and \(Y\) denote the coordinates of a point uniformly chosen in the circle of radius 1 centered at the origin. That is, their joint density is $$f(x, y)=\frac{1}{\pi} \quad x^{2}+y^{2} \leq 1$$ Find the joint density function of the polar coordinates \(R=\left(X^{2}+Y^{2}\right)^{1 / 2}\) and \(\Theta=\tan ^{-1} Y / X\)
Short Answer
Expert verified
The joint density function of the polar coordinates \(R\) and \(\Theta\) is given by:
\[g(R, \Theta) = \frac{R}{\pi}\]
for \(0 \leq R \leq 1\) and \(0 \leq \Theta \leq 2\pi\).
Step by step solution
01
Review the relationship between Cartesian and polar coordinates
Recall that the Cartesian coordinates \(X\) and \(Y\) can be represented in polar coordinates \(R\) and \(\Theta\) as follows:
\[X = R \cos \Theta\]
\[Y = R \sin \Theta\]
Since we are given the joint density function of \(X\) and \(Y\) and we want to find the joint density function of \(R\) and \(\Theta\), we will need to make use of the Jacobian transformation.
02
Use the Jacobian transformation technique
The Jacobian transformation technique is used to find the joint density function of a new set of variables formed from an existing set of variables. The Jacobian transformation formula is given by:
\[g(u, v) = f(x, y) |J(u, v)|\]
where \(J(u, v)\) is the determinant of the Jacobian matrix of the transformation.
First, we need to find the Jacobian matrix of the transformation from Cartesian to polar coordinates.
Let
\[u = R\]
\[v = \Theta\]
The inverse transformation from polar to Cartesian coordinates is given by:
\[x = u \cos v\]
\[y = u \sin v\]
Now, we can find the partial derivatives resulting in the Jacobian matrix:
\[\begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{bmatrix}\]
Computing the partial derivatives, we get:
\[\begin{bmatrix}
\cos v & -u \sin v \\
\sin v & u \cos v
\end{bmatrix}\]
Now, compute the determinant of the Jacobian matrix (\(|J(u, v)|\)):
\[
|J(u,v)| = \left(\cos{v}\right)(u \cos{v}) - \left(-u \sin{v}\right)(\sin{v}) = u
\]
The joint density function of \(X\) and \(Y\) is \(f(x, y) = \frac{1}{\pi}\) when \(x^2 + y^2 \leq 1\), and 0 otherwise. We can now compute the joint density function of \(R\) and \(\Theta\) using the given transformation technique formula.
The transformed joint density function will be:
\[g(u, v) = f(u \cos v, u \sin v) |J(u, v)|\]
For \(x^2 + y^2 \leq 1\), substitute \(x = u\cos{v}\) and \(y = u\sin{v}\):
\[u^2 \leq 1\]
As \(u = R\), this implies that \(0 \leq R \leq 1\).
Now, using the relation
\[g(u, v) = f(u \cos v, u \sin v) |J(u, v)|\]
We get:
\[g(u, v) = \frac{1}{\pi} \cdot u\]
So, the joint density function of \(R\) and \(\Theta\) is given by:
\[g(R, \Theta) = \frac{R}{\pi}\]
for \(0 \leq R \leq 1\) and \(0 \leq \Theta \leq 2\pi\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Joint Density Function
In probability theory, the joint density function describes how two random variables are distributed together. When dealing with continuous random variables, a joint density function is a mathematical function that gives the probability that two random variables fall within a certain range, particularly over a given area in the plane. This is critical for understanding how one variable affects the other.
- The joint density function of two random variables, say, \(X\) and \(Y\), is denoted as \(f(x, y)\).
- This function describes how the probability that \(X\) takes a particular value while simultaneously \(Y\) takes another value is distributed over the plane.
Polar Coordinates
Polar coordinates are a system in mathematics for representing points in a plane using a distance from a reference point and an angle from a reference direction. Unlike the Cartesian coordinate system, which uses two perpendicular axes, polar coordinates use a central point, called the pole (analogous to the origin in Cartesian coordinates), and an angle \(\Theta\) from a reference direction.
- The radius \(R\) represents the distance from the origin to a point.
- The angle \(\Theta\) is measured from the positive x-axis, with angles increasing counterclockwise.
Jacobian Transformation
The Jacobian transformation is a mathematical method used to transform coordinates for multivariable functions. It's particularly helpful when transitioning from one coordinate system to another, such as from Cartesian to polar coordinates in this exercise. The Jacobian is used to adjust the scale of the transformation, allowing us to map one region in one coordinate system to another in a new system while maintaining the probability density.Here are the steps simplified:
- Identify the transformation equations. In our case, the transformation from Cartesian \((x, y)\) to polar \((R, \Theta)\) was used.
- Calculate the Jacobian matrix, which contains partial derivatives of the transformation functions with respect to the original variables.
- Compute the determinant of the Jacobian matrix. This number scales the density function accordingly during the transformation process.
Uniform Distribution
A uniform distribution is a type of probability distribution in which every outcome is equally likely. It can be continuous, where the probability is spread over an interval or a shape, such as a rectangle or circle.In this exercise, the coordinates \(X, Y\) are uniformly distributed over the unit circle. This means that no specific region within the circle is more likely to be chosen than any other. Every point within the circle has an equal chance of being selected.
- The joint density function for a uniform distribution over a continuous area, like a circle, is constant within its bounds and zero outside this region.
- For the unit circle, the density was \( \frac{1}{\pi} \), reflecting uniform probability spread across the circle.
