# Proving the second property of Density Functions

• May 23rd 2009, 06:03 AM
Bucephalus
Proving the second property of Density Functions
hi

I'm trying to prove the four properties of the density functions.

The question goes, "Show that the properties of a density function $f_X(x)$ are valid."

I have done the first one, but I am having trouble with the second one.
The second property of density functions in my books goes like this....

$\int^\infty_{-\infty}f_X(x)dx = 1$

can anyone help?
• May 23rd 2009, 06:50 AM
Laurent
Quote:

Originally Posted by Bucephalus
hi

I'm trying to prove the four properties of the density functions.

The question goes, "Show that the properties of a density function $f_X(x)$ are valid."

I have done the first one, but I am having trouble with the second one.
The second property of density functions in my books goes like this....

$\int^\infty_{-\infty}f_X(x)dx = 1$

can anyone help?

Remember that a probability must satisfy $P(\Omega)=1$ where $\Omega$ is the whole probability space... In your case, $\Omega=\mathbb{R}$.
• May 23rd 2009, 03:05 PM
matheagle
Do you have a specific density?
Or are you asking why a density must integrate to one?
Since probability=area, the total area=total probability=1.
• May 23rd 2009, 03:17 PM
Bucephalus
Where to from here
Thanks,

How do I say this though.
Do I just say that $f_X(x)$ represents the probability space which is 1. Therefore the area under the curve is 1.

I was trying to prove in terms of the integral result which is also $F_X(x)$. A function that extends from $-\infty$ at a value of 0 to $\infty$ at a value of 1.
Is there anything in this function that would suggest that it's derivative.
I think it just dropped...does this sound like a valid.
the function that is the integral of $f_X(x)$ is $F_X(x)$ and if we take the values of $F_X(x)$ at $\infty$ and $-\infty$, i.e. the limits, it will evaluate to 1 - 0 = 1.

Is that a valid validation?
• May 24th 2009, 12:02 AM
Moo
Hello,

$\mathbb{P}(X\in A)=\int_A f_X(x) ~dx$ by definition.

Hence $\int_{-\infty}^\infty f_X(x) ~dx=\mathbb{P}(X\in\mathbb{R})$

But (you didn't state it but we can assume) X has its values in $\mathbb{R}$. So, as Laurent said, $\Omega=\mathbb{R}$

I think that since you're asked to prove the properties of the pdf, you wouldn't be asked to use the cdf...
• May 24th 2009, 02:30 AM
Bucephalus
Thanks to all
Thanks for you help.
It's starting to make sense now.