I’ve been passed a probability density function that I don’t recognize. Worse, I’ve not been given any values for the various constants. I’ve gone digging but can’t seem to identify it so I turn to the Dope.
p(x) = F/2 for x <= T
p(x) = (1-F)(B-1) / X[sub]m[/sub] * (x/X[sub]m[/sub])^(-B) for x>T, where T=2
Any ideas?
Just guess.
As specified, the cdf is not valid because there’s no point at which it’s equal to zero. Was there an additional constrain that p(x) = 0 for x < 0? Regardless, you’re working with a mixed distribution here, so it’s not going to be any of the distributions you’d normally see in textbooks.
ultrafilter is right about the poorly specified domain.
If this pdf is only valid for x>0, then there are a few things you can figure out. First, the distribution is a mixture of a uniform distribution (on [0,T]) and a power-law distribution (on [T,infinity)). You specify T=2, so integrating the uniform distribution gives a probability of (F/2)*T=F. This tells you the interpretation of F: with probability F you’re choosing from the uniform distribution, and otherwise you choose from the power law.
Now integrating the power-law distribution only gives a finite answer for B>1; in this case it integrates to (1-F)(T/X[sub]m[/sub])[sup]1-B[/sup]. We know that it should integrate to 1-F, though, so we must have X[sub]m[/sub]=T=2.
If you did not have T=2, the requirement that the density be normalized would give a more complicated constraint on the four parameters.