TITLE: Expectation of squared time-scaled Brownian process
QUESTION [5 upvotes]: According to an article I'm studying ("Time series, self-similarity
and network traffic by Mark Crovella) the expectation of the square of
a time-scaled Brownian motion process $E[ B(ct)^2 ]$ where $c$ is the time
scaling is equal to $ct$.
I'd appreciate help proving this; i.e.
$E[  B(ct)^2 ] = c t$

REPLY [7 votes]: This follows from the fact that $B(t)$ has a Gaussian$(0,t)$ distribution.  Therefore, $B(ct)$ has a Gaussian$(0,ct)$ distribution.  Thus $E[B(ct)^2] = Var[B(ct)] + (E[B(ct)])^2 = ct + 0 = ct$.