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The exponential distribution and the Poisson distribution.
Yes, and is equal to 1. This is true for normal distribution using any mean and variance.
The mean and variance are equal in the Poisson distribution. The mean and std deviation would be equal only for the case of mean = 1. See related link.
Yes, mode equals median in a normal distribution.
The standard normal distribution is a subset of a normal distribution. It has the properties of mean equal to zero and a standard deviation equal to one. There is only one standard normal distribution and no others so it could be considered the "perfect" one.