Conditional distribution of continuous random variables

Hello, I have a question about conditional distributions in the continuous case.

We are given that the joint distribution function of two continuous random variables X and Y is

for

The marginal distribution of Y can be found by integrating with respect to x, the marginal distribution of Y is

for

The book asks to find the conditional density of X given that Y =.3

This is the answer that is listed in the book

Why is the book just plugging in the value .3 for y? Shouldn't we be integrating since this is continous?

Re: Conditional distribution of continuous random variables

Hey Jame.

You can think of the conditional distribution as the slice (or cross sectional) distribution where the Y value is fixed. Since you knowe the value of y, you treat it like a constant and substitute it in to get the distribution for the random variable X.

If you think of it as a constant and X being the random variable, then you get a PDF in terms of X only with all the properties (integrates to 1, greater than zero, and so on).

Re: Conditional distribution of continuous random variables

That makes sense.

I guess I was thrown off by the single value and the fact that the rvs were continous.

If I was given that Y was between 0 and .3, then would I integrate?

Thank you for assistance.

Re: Conditional distribution of continuous random variables

Yes you would integrate in that situation over the appropriate region.