# Marginal Density of Coordinates Inside an Ellipse

• gajohnson
In summary, the conversation discusses finding the marginal densities of the X and Y coordinates of a point chosen randomly in the interior of an ellipse. The density function for (x,y) is 1/∏ab and the limits of integration are -(b/a)\sqrt{a^2 - x^2} and b/a\sqrt{a^2 - x^2}. The reasoning is deemed correct and the conversation ends with gratitude towards the helpers.
gajohnson

## Homework Statement

A point is chosen randomly in the interior of an ellipse:

(x/a)^2 + (y/b)^2 = 1

Find the marginal densities of the X and Y coordinates of the points.

NA

## The Attempt at a Solution

So this ought to be uniformly distributed, thus the density function for $(x,y)$ is $f_{x,y}$ = $1/∏ab$ (where ∏ab is the area of the ellipse)

So, to find the marginal density for x (and later for y), I realize that I just need to find the limits of integration and then go about my business. I believe that the limits of integration are
$-(b/a)\sqrt{a^2 - x^2}$ and $b/a\sqrt{a^2 - x^2}$,

since these should be the minimum and maximum values that y can take for any given x. Are these limits of integration and/or is my reasoning correct?

As always, many thanks to all of you wonderful Homework Helpers!

Last edited:
That seems entirely correct.

clamtrox said:
That seems entirely correct.

Excellent, thanks!

## 1. What is the marginal density of coordinates inside an ellipse?

The marginal density of coordinates inside an ellipse refers to the probability distribution of points within an ellipse when the coordinates are randomly selected. It is a measure of the likelihood of a point falling within a specific region of an ellipse.

## 2. How is the marginal density of coordinates inside an ellipse calculated?

The marginal density of coordinates inside an ellipse can be calculated using the formula for the probability density function of a bivariate normal distribution. This involves integrating the function over the region of interest, which in this case is the ellipse.

## 3. What factors can affect the marginal density of coordinates inside an ellipse?

The marginal density of coordinates inside an ellipse can be affected by the shape and size of the ellipse, as well as the mean and variance of the bivariate normal distribution. The orientation and position of the ellipse can also have an impact on the marginal density.

## 4. How is the marginal density of coordinates inside an ellipse used in scientific research?

The marginal density of coordinates inside an ellipse can be used in various fields of study, such as statistics, physics, and economics. It can be applied to model and analyze data that follow a normal distribution, or to understand the behavior of particles or systems within an elliptical region.

## 5. Can the marginal density of coordinates inside an ellipse be generalized to other shapes?

Yes, the concept of marginal density can be extended to other shapes and regions. The specific formula and calculations may differ depending on the shape and distribution of the data, but the concept of measuring the probability of points falling within a specific region remains the same.

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