You do not solve a standard normal distribution. It is not a question nor an equation or inequality to be solved. You can answer questions using the standard normal distribution but what you do depends on the question and on what information is given.
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The standard deviation in a standard normal distribution is 1.
The standard deviation in a standard normal distribution is 1.
The standard normal distribution is a special case normal distribution, which has a mean of zero and a standard deviation of one.
Yes, the normal distribution, standard or not is always continuous.
The mean of a standard normal distribution is 0.