The Standard Normal Distribution

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The Standard (0,1) Normal Distribution

Suppose we make the transformation

Z = (X-m)/s

It turns out that by doing so, Z (which is also a random variable) also follows a Normal distribution, but with a mean m=0 and a SD s=1. Thus the pdf of Z is given by

The r.v. Z with the above pdf is called the standard normal random variable (or sometimes, the unit normal random variable). It is an artificial distribution that rarely occurs by itself in practice but it is very useful in dealing with an arbitrary normal distribution.

The cdf of Z, i.e., P(Z£ z) = is denoted by F(z).

z_a Notation: The value z_a represents that 100*(1-a )^th percentile of the standard normal distribution, i.e.,
P(Z£ z_a) = F(z_a) = (1-a), or equivalently, P(Z³ z_a) = a. Thus z_a is the point on the axis for which the area under the pdf curve to the right of it is equal to a.

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Jayant Rajgopal