distribution functions

[buddingscientist]

A continuous random variable X has the cumulative distribution function
F(x) = 0 if x < 1
= 1/3 (sqrt(2x-1)) if 1<= x < 5
= 1 if x >= 5

find the probability Density function

any ideas?

[mathman]

The density function is simply the derivative of F(x). However, because F is discontinuous at x=1, mathematically speaking the density function does not exist. If you a little less fussy, you can have a delta function with weight 1/3 at that point.

Another way of thinking about it, is that the random variable isn't continuous,
i.e. P(X=1)=1/3.

[buddingscientist]

oooh !
So if we were to find the prob X between 2 and 3, we would simply sum f(2) and f(3) and not do any integral stuff.
(Because it's discrete and only has integer values)
thanks very much

[mathman]

The random variable has an "atom" (i.e. a point with P>0) at 1. It has continuous probability from 1 to 5, and 0 probability outside. If this is a homework problem, it sounds a little messy. Are you sure you've got the definition of F(x) correct?

[buddingscientist]

Hi
This was a past class test question, and unfortunatly there are no answers so one is only left to wonder whether they know how to do these sorts of problems or not.

The test is over and I don't think I answered the question to do with cumu. dist. functions correctly btu we can only wait and see next thurs