Consider an Avalanche PhotoDiode, an APD: we set up an exotic state of matter so that the output signal is almost always near zero current, but occasionally it is some obviously non-zero value. Hardware is usually set up to record the time at which a transition from zero to non-zero current happens (we could instead record the current as a 14-bit output from an Analog-to-Digital Converter, an ADC, every nanosecond, say, but the record of current transition times is essentially a very compressed, very lossy record of the same information.) Also of interest in experiments is the dead time, the time it takes the hardware to restore the current to near zero so that another transition can be noticed and the time recorded.
Suppose we have this device. When it's set up in a dark room, there is a low rate of current transitions, called the dark rate; when we enter the room and turn on a dim light, the rate of current transitions changes; when we move around the room, the rate of current transitions changes; when we change the intensity of the light or introduce new lights, the rate of current transitions changes. If we set up some barriers, again the rate of current transitions changes, and again when we move the barriers around. If we set up two or more APDs, we can calculate more elaborate statistics, cross-correlations at the same or at different times.
If we ask what could be causing these events, one answer is that we've set up a ridiculously exotic state of matter, so of course weird stuff will happen. More than that, however, we notice that as we continuously change the conditions of the experiment, the current transition statistics change more-or-less continuously, if we collect enough data. Even though the events are discrete, the statistics change continuously. Historically, elementary physics has said that each current transition is caused by a particle, but more sophisticated physics works with a quantum
field, which can be understood to make no claims about what happens outside the APD, nor about details of the APD current, but does discuss the statistics one would observe for a given theoretical model of an APD, and how those statistics would change continuously as we move the lights or the barriers or the APDs around.
For what it's worth, my YouTube video from last February,
Quantum Mechanics: Event Thinking, deliberately short at 4'26", presents more-or-less this story.