Uniform distribution of a disc

AI Thread Summary
The discussion revolves around finding the marginal probability density functions (pdf) for a uniform distribution over a disc of radius 1. The joint pdf is established as 1/π for points within the disc defined by x² + y² ≤ 1. To derive the marginal pdf of x, one must integrate the joint pdf over the appropriate limits for y. The correct limits for integration are determined to be from -√(1 - x²) to √(1 - x²), reflecting the possible y-values for a fixed x within the unit circle. The conversation emphasizes the importance of visualizing the unit circle to accurately set the integration boundaries.
mathmathmad
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Homework Statement



Consider a disc of radius 1 in the plane D in R
D = {(x,y) in R | x^2+y^2 <= 1}
write the marginal pdf of x and y

Homework Equations


The Attempt at a Solution


so the joint pdf is 1/Pi for x^2 + y^2 <= 1 <- correct?
but how to I get the marginal pdfs?
 
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You have

<br /> f(x,y) = \frac 1 \pi<br />
as the joint density. As in every case the marginal density of one variable is found by integrating out the other. Suppose you want the marginal density of x - you need to integrate out y.

<br /> g(x) = \int_a^b \frac 1 \pi \, dy<br />

The question is: what should you use for the limits a and b?
Draw the unit circle and see the set of y-values that are possible when x is fixed.
 
a=0
b=sqrt (1 - x^2) ?
 
mathmathmad said:
a=0
b=sqrt (1 - x^2) ?

Try again. Did you draw the picture?
 
a = - sqrt (1 - x^2)
b = sqrt (1 - x^2)

or should it be -1 to 1
 
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