Thread: Conditional density function?

Hi, I'm not sure what the notation is supposed to mean in the highlighted box and how it relates to conditional probability:
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The definition of conditional probability that I'm familiar with is P(A|B) = P(A, B)/P(B). How (if at all) does that relate to what the authors have written there? Some special properties of the normal distribution perhaps?

PS. Theta is a vector of reals, as is x, so

Firstly, check that you understand how this relates to the previous line. It is simply making the subsititution


The notation appears to be a conditional pdf for Y, ie (very loosely*):



That is to say, in very loose terms* it is telling your the conditional probability of Y|X, if the parameter value is equal to theta. The parameter value has to be included because it made the substitution , which will be different for every possible value of


*in fact, pdfs do not give probabilities, but thats a seperate topic...

Thanks a lot! But didn't they make the substitution eps = y - theta . x instead? Since the mean of the normal dist is theta . x ?

yeh i just typoed the subsitituion formula


Should have been

which rearranges to what you said

But then I suppose my question is, if they made that substitution for , why did they change

to , and not or something like that (if that even makes sense)? I.e. Why did it become conditional all of a sudden?

i think:
Remember that once you condition on x, is just a number. This means you can make the substitution without worrying about how the randomness of x will affect the distribution of y.