The probability that the variate would take is denoted by the shaded area in the figure.The variate would take a value between 0 and z. This can be read from the table of areas under Standard Normal Curve. The area from 0 to z can be found to any corresponding z from this table. Let X is a normal variate with mean µ and standard deviation σ. Then Z=(X-µ)/σ is a Standard Normal Variate.Hence the Standard Normal Variate can be used to find the probability regarding X.
The expected value of a standard normal variable X is E[X]=0 and the variance is Var[X]=1. The characteristic function of a standard normal random variable X is: ϕX(t) = exp((-1/2)t2).
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