What do the realizations of X(t)=Usin(t)+Vcos(t) where U and V are random variables with mean 0 and and variance 1 look like from -2pi to 2pi? I'm not sure what the realizations of a time series really mean, and how to implement any kind of drawing with random variables. Any hints or descriptions would be very helpful.  
 A: A random process is a collection of random variables 
$\{X(t)\colon t \in \mathbb T\}$, one variable $X(t)$ for each time instant $t$ in the index set $\mathbb T$.  Typically, $\mathbb T$ is the set
of all real numbers $\mathbb R$, 
or the set of all positive real numbers $\mathbb R^+$,
or the set of all integers $\mathbb Z$, etc. All of these random variables are defined on a common sample space $\Omega$, that is, each is a  mapping from
$\Omega$ to the real line. Thus when the experiment is performed and 
the outcome $\omega \in \Omega$ is known, 
each $X(t)$ maps that outcome $\omega$ onto its corresponding
real number which we can call $x(t)$ for convenience. Thus, the
collection of numbers $\{x(t)\colon t \in \mathbb T\}$ is
the realization or sample path of the process corresponding
to that particular outcome $\omega \in \Omega$.
In your example where $X(t) = U\cos t + V \sin t$, (and
presumably $\mathbb T = \mathbb R$), many of the
random variables are identical: $X(t)$ is the same random
variable as $X(t+2n\pi)$ for all integers $n$. When the experiment
is performed, $U$ and $V$ take on values $u$ and $v$, say,
and thus
$$x(t) = u\cos t + v\sin t 
= \sqrt{u^2+v^2}\cos\left(t - \arctan\left(\frac vu\right)\right)$$
is a sinusoidal function of period $2\pi$.
