Yes.
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None. The full name is the Probability Distribution Function (pdf).
They are the same. The full name is the Probability Distribution Function (pdf).
A probability density function assigns a probability value for each point in the domain of the random variable. The probability distribution assigns the same probability to subsets of that domain.
No. f is a letter of the Roman alphabet. It cannot be a probability density function.
It is a function which is usually used with continuous distributions, to give the probability associated with different values of the variable.