The Standard Normal Distribution

Introduces the standard normal distribution $N(0,1)$ and its cumulative distribution function $\Phi$ . Covers reading $P(Z \leq a)$ directly from a standard normal table, the complement rule $P(Z > a) = 1 - \Phi(a)$ , the symmetry identity $\Phi(-a) = 1 - \Phi(a)$ , and combining these to compute $P(a \leq Z \leq b)$ .

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Introduction

The standard normal distribution is a continuous probability distribution with mean $\mu = 0$ and standard deviation $\sigma = 1$ . A random variable $Z$ following this distribution is written $Z \sim N(0, 1)$ .

Its probability density function is

\varphi(z) = \dfrac{1}{\sqrt{2\pi}}\, e^{-z^2/2},

whose graph is the familiar bell-shaped curve: symmetric about $z = 0$ , with total area under the curve equal to $1$ .

Probabilities involving $Z$ correspond to areas under this curve. The cumulative distribution function of $Z$ is

\Phi(a) = P(Z \leq a).

Values of $\Phi(a)$ for $a \geq 0$ are listed in a standard normal table; for example, $\Phi(1.00) = 0.8413$ .

Two facts follow immediately from the shape of the curve:

By symmetry about $z = 0$ , exactly half the area lies to the left of $0$ , so $\Phi(0) = 0.5$ .
Because $Z$ is continuous, $P(Z = a) = 0$ for every $a$ . Therefore $P(Z < a) = P(Z \leq a) = \Phi(a)$ .