10D

The standard normal distribution

The standard normal distribution with mean 0 and standard deviation 1, its bell-shaped PDF, finding probabilities with a table or calculator, symmetry, and quartiles.

The bell curve

Standard normal PDF

$$\phi(z) = \frac{1}{\sqrt{2\pi}}\,e^{-\frac12 z^2}$$

The standard normal variable $Z$ has mean $0$ and standard deviation $1$. Its curve is bell-shaped, symmetric about $z = 0$, has its maximum at $z = 0$, and gets very close to the axis beyond $z = \pm 3$. The total area is $1$.

Areas under the standard normal curve

Slide a. At a = 0 the shaded area is exactly one half, by symmetry.

Try this

At $a = 1$ the shaded area is about $0.8413$. Where is the remaining $0.1587$?

Finding probabilities

There is no simple primitive for $\phi$, so use a table or calculator for $P(Z \le a)$. The table only gives areas to the left, so use these facts for everything else.

$a$	0.5	1	1.5	2
$P(Z \le a)$	0.6915	0.8413	0.9332	0.9772

Example 1●

Find $P(Z \gt 1)$ and $P(Z \lt -1)$.

$P(Z \gt 1) = 1 - 0.8413 = 0.1587$.

By symmetry, $P(Z \lt -1) = P(Z \gt 1) = 0.1587$.

Example 2●●

Find $P(-1 \lt Z \lt 2)$.

$P(Z \lt 2) - P(Z \lt -1) = 0.9772 - 0.1587 = 0.8185$.

Example 3: working backwards●●●

Find the upper quartile of $Z$.

We need $P(Z \le q) = 0.75$. A table or calculator gives $q \approx 0.674$.

By symmetry the lower quartile is $-0.674$.

Flashcards

Mean and sd of $Z$

$0$ and $1$

$P(Z gt a)$

$1 - P(Z \le a)$

$P(Z le -a)$

$P(Z \ge a)$

$P(Z le 0)$

$0.5$

Practice

Find $P(Z \gt 1)$, given $P(Z \le 1) = 0.8413$.
Find $P(Z \lt -1)$.
Find $P(-1 \lt Z \lt 2)$, given $P(Z \le 2) = 0.9772$.
Find $P(-1 \lt Z \lt 1)$.
Find $P(Z \gt 1.5)$, given $P(Z \le 1.5) = 0.9332$.
Find $P(Z \gt 0)$.
Find $P(Z \lt -0.5)$, given $P(Z \le 0.5) = 0.6915$.
The standard normal curve has its maximum at:
The lower quartile of $Z$ is about: