Finding the average value of $\frac{1}{N}\sum_{k=1}^{N}{\cos(2 \pi f_0 (1+y_k)t)}$ where $y_k$ is a small random number The problem is

Find an average value of $$\frac{1}{N}\sum_{k=1}^{N}{\cos(2 \pi f_0 (1+y_k)t)},$$ where $y_k$ is picked randomly from the interval $\left[-\frac{\Delta f}{2f_0},\frac{\Delta f}{2f_0}\right]$.

$f_0$ is the center frequency and the distribution from which various $y_k$ are picked can be a Gaussian or uniform distribution, whichever is more simple for the purpose of the explanation. The width of the distribution is $\Delta f$. $t$ is just some parameter (like time which can take any value from $0$ to $\infty$)
Note that we want the average of an average of random cosines.
My first thought was to expand using $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ identity. However, $\sin(b)$ term cannot be neglected as $2\pi f_0 t$ factor can be as large as possible.
I have literally no idea how to go about solving it because $y_k$ is a random variable.
 A: It is customary to designate random variables with uppercase letters, so we will replace the $y_k$ notation in the Question with $Y_k$, reserving lowercase for ordinary (real) values below.
Any problem asking for the "average value" of an expression involving a random variable must specify the probability distribution of the values it takes.  Here assume all the $Y_k$ random variables have a continuous probability density $p(y)$ on the common interval $[-\Delta f/2f_0,+\Delta f/2f_0]$.
The the "average value" of the summation is also known as the expected value:
$$ \mathbb E\left(\frac{1}{N}\sum_{k=1}^{N}{\cos(2 \pi f_0 (1+Y_k)t)}\right) $$
Given the probability density function $p(y)$ one compute the expected value by integrating:
$$ \int_{-\Delta f/2f_0}^{+\Delta f/2f_0} \frac{1}{N}\sum_{k=1}^{N}{\cos(2 \pi f_0 (1+y)t)} p(y) dy $$
Since the sum has finitely many terms, the order of integration and summation can be swapped (linearity of expectation).  But in this case the interval of integration, the probability distribution, and the integrand are all the same, so dividing the sum by $N$ just gives a single term to integrate:
$$ \int_{-\Delta f/2f_0}^{+\Delta f/2f_0} \cos(2 \pi f_0 (1+y)t) p(y) dy $$
To evaluate the integral requires one to supply the desired density function $p(y)$.  The simplest case is a uniform distribution on $[-\Delta f/2f_0,+\Delta f/2f_0]$, so that $p(y)$ is a constant function:
$$ p(y) = \frac{f_0}{\Delta f} $$
Now a $u$-substitution $u = 2\pi f_0 (1+y)t$ makes the integration a routine exercise.
