Is it possible to calculate the expected value of $\sin^2(0.01n)$, with $n$ taking non-negative integer values?
Normally, when we wish to find the expected value of the sine function, we integrate (or add) its corresponding values within a single period (I think?), and then take its mean. Is such a method possible for the function under question?
The phrase "expected value" has no meaning when applied to a sequence $\mathbf x=(x_n)_{n\geqslant1}$, but one can look at the Cesàro means $(c_n(\mathbf x))_{n\geqslant1}$ and at their limit $\mathcal C(\mathbf x)$ defined, if it exists, as $$\mathcal C(\mathbf x)=\lim_{n\to\infty}c_n(\mathbf x)$$ where, for every $n\geqslant1$, $$c_n(\mathbf x)=\frac1n\sum_{k=1}^nx_k$$ In the present case, $x_n=\sin^2(an)$ with $a=0.01$ hence $2x_n=1-\cos(2an)=1-\Re(\mathrm e^{\mathrm ian})$ and $$2c_n(\mathbf x)=1-\frac1n\Re\left(\sum_{k=1}^n\mathrm e^{\mathrm iak}\right)$$ The sum in the RHS is geometric with ratio $z=\mathrm e^{\mathrm ia}$. Thus, the sums in the RHS are bounded since $|z|=1$ and $z\ne1$, hence the sequence $(c_n(\mathbf x))_{n\geqslant1}$ converges and its limit is $$\mathcal C(\mathbf x)=\frac12$$