Cdf (Cumulative Density Function) Confusion

[JoshMaths]

Hi there,

So regular i thought that the procedure was
F(s) = ∫^s₀ f(x) dx

However i am doing a problem with a kinked pdf and it is telling me to do something like

F(s) = ∫^s₀ f(s) ds for 0=<s>=1/2

then...
F(s) = F(1/2) + ∫^s_1/2 f(s) ds

I am confused at the process of using f(x) or f(s) in the integral and the situation with kinked pdfs, if someone can shed some light on this i would be greatful.

Thanks,
Josh

 

[Office_Shredder]

I don't know what the definition of a kinked pdf is, but a couple of corrections -

The usual definition is
$$F(s) = \int_{-\infty}^{s} f(x) dx$$
Your definition only works if you know $f(x) = 0$ for all x<0

After that you have s as a variable of integration and as a bound of integration - be careful not to do that!

The last equation you wrote though is just an application of the fundamental theorem of calculus. Once you have F(s) defined either by your integral or mine, we get that
$$F'(s) = f(s)$$
And therefore for all a and b
$$F(b) - F(a) = \int_{a}^{b} f(x) dx$$
In particular if a = 1/2 and b=s we get
$$F(s) - F(1/2) = \int_{1/2}^{s} f(x) dx$$
Adding F(1/2) to both sides gives your final equation

 

[JoshMaths]

In my university module they define a kinked pdf as a pdf having two functions over differing ranges.

I agree with everything you put here thanks. If you have a look at my solution paper you see that the s variable is part of the integration range and also the function, does this mean that they are wrong?

 

[Office_Shredder]

Using s to mean two different things in the same equation is usually frowned upon - I would change it so the variable inside the integral is different.

 

[JoshMaths]

No i agree i was just confused as these are university lecturers writing this, they shouldn't really be giving bad practices.