How to Find the PDF for a Uniform Distribution on a Disc?

In summary, the pdf for a uniform distribution on a disc of radius 1 is f_{xy} = \frac{(x^2 + y^2)}{\pi} for x^2 + y^2 \leq 1 and 0 otherwise. To find the marginal distribution of x and y, you would integrate f_{xy} over the appropriate limits, which would be the boundaries of the disc.
  • #1
rosh300
17
0

Homework Statement



[tex]\D = \{(x,y) \in \mathbb{R}^2 | x^2 + y^2 \leq 1\} [/tex] i.e. a disc or radius 1.
Write down the pdf f_{xy} for a uniform distribution on the disc.

Homework Equations





The Attempt at a Solution



[tex] f_{xy} = \frac{(x^2 + y^2)}{\pi} \mbox{for} x^2 + y^2 \leq 1[/tex] 0 otherwise
as the area of the disc pi and to make it uniform you divide by pi so the probability integrates to 1
 
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  • #2
Hmmm...

[tex] f_{xy} = \frac{(x^2 + y^2)}{\pi}[/tex]

Doesn't look very uniform to me:wink:
 
  • #3
i think i got it: its [tex]
f(x,y)_{xy} = \left\{ \begin{array}{rl}
\frac{1}{\pi} &\mbox{for } x^2 + y^2 \leq 1\\
0 &\mbox{otherwise}
[/tex]

thanks
 
Last edited:
  • #4
Looks good to me!:approve:
 
  • #5
I am doing a some practice questions for stats and i tried to integrate this to get 1 but i can't so what are the appropriate limits and how would i go about finding the marginal distribution of x and y? Thanks
 

Related to How to Find the PDF for a Uniform Distribution on a Disc?

1. What is a uniform distribution of a disc?

A uniform distribution of a disc refers to a probability distribution where all points on a disc have an equal chance of being selected. This means that the probability of selecting any point on the disc is the same as any other point.

2. How does a uniform distribution of a disc differ from other distributions?

Unlike other distributions, such as a normal or Gaussian distribution, a uniform distribution of a disc does not have a central tendency or peak. Instead, all points on the disc have an equal likelihood of being selected.

3. What is the mathematical formula for a uniform distribution of a disc?

The mathematical formula for a uniform distribution of a disc is P(x) = 1/A, where P(x) is the probability of selecting a point on the disc and A is the area of the disc.

4. How is a uniform distribution of a disc used in scientific research?

A uniform distribution of a disc is often used in scientific research to model random processes, such as the distribution of particles in a solution or the locations of stars in a galaxy. It can also be used to generate random numbers for simulations and experiments.

5. Are there any real-life examples of a uniform distribution of a disc?

Yes, there are many real-life examples of a uniform distribution of a disc. Some examples include the distribution of tree canopies in a forest, the distribution of bacteria on a petri dish, and the distribution of molecules in a gas. In these cases, the disc represents the space in which the objects are distributed.

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