1. ## Mean and Variance

Given the pdf and cdf of a continuous random variable, say X, describe how you would find the mean and variance of X without using integration.

2. Well you could use the $MGF$ but I guess that is defined by integration...

3. Oh wait! I remember how to do this:

Here is an example:
$X$~ $exp(\lambda)$

Differentiating the density identity (ofcourse I'm assuming we can swap the order of differentiation and integration).

$\frac{d}{d\lambda} \int_0^{\infty}\lambda e^{-\lambda x}dx = 1$

$\Rightarrow \int_0^{\infty} (-\lambda x +1) e^{-\lambda x}dx = 0$

Which simplifies to

$\int_0^{\infty} \lambda x e^{-\lambda x} = \int_0^{\infty} xe^{-\lambda x} = \frac{1}{\lambda} = E[X]$

Where the last equality holds by an identity

4. Thank you for the response, but I'm not quite following. Can you use the following as an example and find the mean using the method you described using no integration:

pdf: fx(X) = ((x-5)^3)/4 for 5 <= x <= 7 and 0 elsewhere
cdf: Fx(X) = ((x-5)^4)/16 FOR 5 <= x <= 7

Thanks

5. Start like this

$\frac{d}{dx}\int_5^7 \frac{x^3 - 15x^2 + 75x - 125}{4} dx = \frac{d}{dx}1$

$\Rightarrow \int_5^7 \frac{3x^2-30x+75}{4} = 0$

Um... gimme a sec.