What information does cloud fraction provide about weather on a certain day?
Do low-level (1000-700 hPa) cloud fraction, optical thickness of all clouds and total cloud fraction provide any information about whether it rains or not?
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Depending on context, cloud fraction can mean:
the fraction of the visible sky covered by clouds (total cloud fraction). This is what an observer from below or above sees and this is what is normally reported by measurements
the fraction of a particular area and altitude covered by clouds. This is what is typically reported by models.
The differing definitions need to be kept in mind when comparing models to measurements.
Under normal circumstances, clouds are a necessary but not a sufficient requirement for precipitation (there are some exceptions).
To answer your titular question: No, a cloud fraction of 0 for low clouds does not mean it was raining. Nor does it necessarily mean it was not raining. Remember: a total cloud fraction of 0 means a clear sky, a total cloud fraction of 1 means a completely covered sky. So a low level cloud fraction of 0 means there are no low level clouds.
To answer the question you ask in the content of your post: Statistically speaking, yes. The quantities low-level cloud fraction, cloud optical thickness, and total cloud fraction do contain information about precipitation.
If the total cloud fraction is zero, there is no precipitation.
If the cloud fraction for low clouds is zero, but for mid-level or high clouds is non-zero, it may or may not be raining.
Cloud optical thickness should correlate with precipitation occurrence and intensity.
However, these quantities do not provide complete information about precipitation. You can't say with full certainty that a day is rainy based on these quantities alone — but you can say which days certainly aren't.