Standard Normal Distribution

Standard Normal

The standard normal $Z \sim N(0, 1)$ has:

• Mean $\mu = 0$
• Variance $\sigma^2 = 1$
• PDF: $\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$
• CDF: $\Phi(z) = P(Z \leq z) = \int_{-\infty}^{z} \phi(t) \, dt$

Why Standardise?

No closed form exists for $\Phi(z)$. We use tables or software to find $\Phi(z)$ for specific $z$.

Tables give $\Phi(z)$ for positive $z$. By symmetry:

$\Phi(-z) = 1 - \Phi(z)$

Common Values

$z$	$\Phi(z)$
0.0	0.5000
0.5	0.6915
1.0	0.8413
1.5	0.9332
1.645	0.9500
1.96	0.9750
2.0	0.9772
2.5	0.9938
3.0	0.9987

Using Tables

Finding $P(Z \leq z)$ for $z > 0$

Look up $z$ in the table.

Finding $P(Z \leq z)$ for $z < 0$

Use symmetry: $\Phi(-|z|) = 1 - \Phi(|z|)$.

Example: $P(Z \leq -1.5) = 1 - P(Z \leq 1.5) = 1 - 0.9332 = 0.0668$.

Finding $P(a \leq Z \leq b)$

$P(a \leq Z \leq b) = \Phi(b) - \Phi(a)$

Example: $P(-1 \leq Z \leq 2) = \Phi(2) - \Phi(-1) = 0.9772 - 0.1587 = 0.8185$.

Inverse: Finding Quantiles

Given $\Phi(z_p) = p$, find $z_p$.

For $p > 0.5$: Look up the value directly.

For $p < 0.5$: Use symmetry: $z_p = -z_{1-p}$.

Critical Values

Probability	$z$ value
90%	$\pm 1.645$
95%	$\pm 1.96$
99%	$\pm 2.576$

These are central interval boundaries.

Standardisation

For $X \sim N(\mu, \sigma^2)$:

1. Convert $X$ to $Z$: $Z = \frac{X - \mu}{\sigma}$
2. Look up $\Phi(z)$
3. Interpret the result

Example: Battery Lifetime

Batteries have mean lifetime 500 hours, SD 20 hours, normally distributed. What proportion last over 530 hours?

$P(X > 530) = P\left(Z > \frac{530 - 500}{20}\right) = P(Z > 1.5)$

$= 1 - \Phi(1.5) = 1 - 0.9332 = 0.0668$

About 6.68%.

Example: Within k Standard Deviations

What proportion of values lie within 1.5 standard deviations of the mean?

$P(|Z| < 1.5) = P(-1.5 < Z < 1.5) = \Phi(1.5) - \Phi(-1.5)$

$= \Phi(1.5) - (1 - \Phi(1.5)) = 2\Phi(1.5) - 1$

$= 2(0.9332) - 1 = 0.8664$

About 86.6%.

PDF Properties

Symmetry

$\phi(-z) = \phi(z)$ $\phi(z) \text{ maximised at } z = 0$ $\phi(0) = \frac{1}{\sqrt{2\pi}} \approx 0.399$

Asymptotic

$\lim_{z \to \pm\infty} \phi(z) = 0$

The tails approach zero but never reach it.

Summary

Task	Method
$P(Z \leq z)$, $z > 0$	Look up in table
$P(Z \leq z)$, $z < 0$	$1 - \Phi(
$P(a \leq Z \leq b)$	$\Phi(b) - \Phi(a)$
$P(Z > z)$	$1 - \Phi(z)$
Find $z_p$ from $p$	Inverse lookup or symmetry