How can a signal be both periodic and random?
An example you run typically across in a text book (Papoulis as an example) is the sine with random phase
$$
x(t)=\sin(2\pi f + \phi)
$$
where $\phi$ is a random variable, distributed uniformly, over $0$ to $2\pi$.  

Any realization will have $\phi$ take on a particular value, but it’s random, just like a 6 on a dice after a throw. You could not predict it would be 6 before the throw or in this case you could not predict $\phi$ before you received the signal. 

At the other extreme, you can construct a very “random” looking waveform using something like randn() in Matlab. It will pass several statistical tests for randomness but if you use the same seed, the waveform is perfectly repeatable (deterministic). 

Every waveform has a story ( a context ) and the story is how you judge if something is random or not.