The exponential distribution and the Poisson distribution.
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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.
Divide the total number of incidents by the total time. The result, representing the average number of incidents per unit of time, is the mean as well as the variance of the Poisson distribution.
Yes, and is equal to 1. This is true for normal distribution using any mean and variance.
A normal distribution can have any value for its mean and any positive value for its variance. A standard normal distribution has mean 0 and variance 1.
It is exp(20t + 25/2*t^2).