# Thread: Conditional Distribution over the unit Disk

1. ## Conditional Distribution over the unit Disk

Suppose the Random Variables and are uniform (-1,1). How can I show that their conditional distribution, given , is

Can this be derived using the conditional distribution quotient? I would appreciate it if someone could explain all necessary steps.

Thank you.

2. ## Re: Conditional Distribution over the unit Disk

I suppose are given as independent.

First of all you will need to derive the conditional joint CDF, then differentiate it to obtain the conditional joint pdf.

i.e. to calculate the following first:

[math] F_{U,V|U^2+V^2<1}(u,v) [/math

Now the remaining part is just a calculus problem. Sketch the region in your mind

When , the above double integral can be expressed as

and by Fundamental Theorem of Calculus,

When , we have an additional extra term

but since it is independent of , the derivative vanish.

Therefore the result follows.

3. ## The Following User Says Thank You to BGM For This Useful Post:

JohnK (01-14-2014)

4. ## Re: Conditional Distribution over the unit Disk

Thank you but can you please explain the integrals a bit? I'm having some trouble comprehending the expressions and their limits.

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