Changing the Gaussian Distribution from cartesian to polar coordinates

In summary, you need to show that the constant A in the Gaussian distribution P(x)=Ae^(-Bx^2) is equal to sqrt(B/Pi). To do this, you need to insist that the sum over probabilities must equal unity by integrating the function. To make this integral easier, you can first square it and then combine the integrands. By turning the area integral over x and y into an area integral over polar coordinates, you can simplify the calculation and arrive at the final answer of 1=A^2Integral(e^(-B(x^2+y^2)dxdy)=A^2Integral((e^-Br^2)(r)drdθ)=PiA^2/B. If you are struggling
  • #1
stepheckert
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Homework Statement


"You are now going to show that, in the Gaussian distribution P(x)=Ae^(-Bx^2) the constant A is equal to sqrt(B/Pi). Do this by insisting that the sum over probabilities must equal unity, Integral(P(x)dx)=1. To make this difficult integral easier, frst square it then combine the integrands and turn the area integral, over x and y into an area integral over polar coordinates.


The Attempt at a Solution


The back of the book has this answer:
1=A^2Integral(e^(-B(x^2+y^2)dxdy)=A^2Integral((e^-Br^2)(r)drdθ)=PiA^2/B.

I understand the first piece but I don't understand how to get from these cartesian to polar coordinates at all, and I'm very confused as to how they got the final answer from the last integral.

Please Help!
Thanks!
 
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  • #2
You should review changing coordinates, which was covered in your calculus course.
 

1. What is the difference between cartesian and polar coordinates?

Cartesian coordinates use x and y axes to plot points on a two-dimensional plane, while polar coordinates use a radius and angle to describe the position of a point in a two-dimensional space.

2. How do you convert from cartesian to polar coordinates?

To convert from cartesian to polar coordinates, you can use the following formulas:
r = √(x^2 + y^2)
θ = tan^-1 (y/x)
where r is the radius and θ is the angle in radians.

3. Can any Gaussian distribution be converted from cartesian to polar coordinates?

No, only Gaussian distributions that are symmetric about the origin (0,0) can be converted from cartesian to polar coordinates. This means that the distribution must have the same shape and spread on both sides of the origin.

4. Why would one want to convert a Gaussian distribution from cartesian to polar coordinates?

Converting a Gaussian distribution from cartesian to polar coordinates can sometimes make it easier to analyze or model certain data. For example, it may be easier to visualize the data or to identify patterns or relationships using polar coordinates.

5. Does converting from cartesian to polar coordinates change the shape of the Gaussian distribution?

No, converting from cartesian to polar coordinates does not change the shape of the Gaussian distribution. It simply represents the same data in a different coordinate system.

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