Probability density function of transformed random variable

[mnb96]

Hello,
given a continuous random variable x with a known PDF, how can we determine in general the PDF of the transformed variable f(x) ?
For example f(x)=x+1, of f(x)=x² ... ?

Also, if we have two random variables x,y and their PDF's, is it always impossible to determine the PDF of f(x,y), unless we known the joint PDF $f_{x,y}(x_0,y_0)$ ?

Thanks.

 

[chiro]

Hello,
given a continuous random variable x with a known PDF, how can we determine in general the PDF of the transformed variable f(x) ?
For example f(x)=x+1, of f(x)=x² ... ?

Also, if we have two random variables x,y and their PDF's, is it always impossible to determine the PDF of f(x,y), unless we known the joint PDF $f_{x,y}(x_0,y_0)$ ?

Thanks.

I'm not sure about an absolute generic version of generating a PDF in any circumstance, but there are methods for specific methods.

For the kind of simple functions you are talking about, you typically have a function and some information on range, and then using the properties of a pdf (integral/sum of pdf over domain = 1), then the pdf (along with the range) can be calculated.

In the case where you are trying to find the pdf of a system that fits

P(X(0) + X(1) + X(2) + X(3) ... + X(N) <= Z)

If the distributions are the same, then something like moment generating functions can be used to derive the distribution of two random variables of the same type and then an inductive argument can be used.

If they are not the same, you can use convolution to find the cumulative distribution function where the random variables are arbitrary.

Also there are assumptions about whether the random variables are iid. If they are not then its usually more complicated.
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I think though for the type of problems you're thinking of like say a transformed pdf (ie f(x)) then you need to have information about the range. Using that its not too hard to get the pdf since the integral over the range is 1.

With the regards to requiring the joint pdf, that's pretty much correct for the generalized case. A lot of cases use independence, but anywhere where there isn't independence then a joint pdf is vital.