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Yes. When we refer to the normal distribution, we are referring to a probability distribution. When we specify the equation of a continuous distribution, such as the normal distribution, we refer to the equation as a probability density function.
Yes, it is.
It means that the probability distribution function of the variable is the Gaussian or normal distribution.
There are a number of differences. The scientific calculator has function like tables and memory to store calculation, while a normal calculator can only do the basic calculations.
Almost all statistical distribution have a mean. It is the expected value of the random variable which is distributed according to that function.