# Thread: Proving the second property of Density Functions

1. ## 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 $\displaystyle f_X(x)$ 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....

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

can anyone help?

2. 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 $\displaystyle f_X(x)$ 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....

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

can anyone help?
Remember that a probability must satisfy $\displaystyle P(\Omega)=1$ where $\displaystyle \Omega$ is the whole probability space... In your case, $\displaystyle \Omega=\mathbb{R}$.

3. 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.

4. ## Where to from here

Thanks,

How do I say this though.
Do I just say that $\displaystyle f_X(x)$ 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$\displaystyle F_X(x)$. A function that extends from $\displaystyle -\infty$ at a value of 0 to $\displaystyle \infty$ 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 $\displaystyle f_X(x)$ is $\displaystyle F_X(x)$ and if we take the values of $\displaystyle F_X(x)$ at $\displaystyle \infty$ and $\displaystyle -\infty$, i.e. the limits, it will evaluate to 1 - 0 = 1.

Is that a valid validation?

5. Hello,

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

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

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

I think that since you're asked to prove the properties of the pdf, you wouldn't be asked to use the cdf...

6. ## Thanks to all

Thanks for you help.
It's starting to make sense now.