Average of cosine waves

Question

What looks like the graph of the function $A=E[cos(\omega t)]$, where $E$ means "average" and $\omega$ is a random variable?

Answer 1

Assuming that $\omega$ is uniformly distributed in range $[a,b]$, we have

$$E[\cos(\omega t)]=\frac1{b-a}\int_a^b\cos(\omega t)d\omega=\frac{\sin(bt)-\sin(at)}{(b-a)t}.$$

For example, with $a=1,b=3.14$:

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In the case of a Gaussian, another Gaussian.