Mathematics

Normal distribution

Properties of the normal curve, standardization of the Z variable, probability calculation and standard normal table usage.

A continuous random variable,

\mathbf{X}

, that follows a normal distribution with mean

\mathbf{\mu}

and standard deviation

\mathbf{\sigma}

, and is designated by

\mathbf{N(\mu, \sigma)}

, has the form:

A continuous random variable,

\mathbf{X}

, that follows a normal distribution with mean

\mathbf{\mu}

and standard deviation

\mathbf{\sigma}

, and is designated by

\mathbf{N(\mu, \sigma)}

, has the form:

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

It has the following properties:

The domain is the set of all real numbers $\mathbb{R}$ .
It is symmetric with respect to the mean $\mu$ .
It has a maximum at the mean $\mu$ .
It increases up to the mean $\mu$ and decreases afterwards.
At the points $\mu - \sigma$ and $\mu + \sigma$ it presents inflection points.
The x-axis is an asymptote of the curve.
The area enclosed by the function and the x-axis is equal to 1.
Being symmetric with respect to the axis passing through $x = \mu$ , it leaves an area equal to $0.5$ to the left and another equal to $0.5$ to the right.
The probability is equivalent to the area enclosed under the curve.
Probability of some intervals:

\begin{aligned} P(\mu - \sigma < X \le \mu + \sigma) &= 0.6826 = 68.26\% \\ P(\mu - 2\sigma < X \le \mu + 2\sigma) &= 0.954 = 95.4\% \\ P(\mu - 3\sigma < X \le \mu + 3\sigma) &= 0.997 = 99.7\% \end{aligned}

Standard normal distribution
In a distribution with mean 0 and standard deviation 1.

\mathbf{N(0, 1)}

. The variable

\mathbf{Z}

is used.

Standardization of a normal distribution: To be able to calculate probabilities in a non-standard distribution,

N(\mu, \sigma)

, of variable

X

, we must transform the variable

X

into another variable that follows an

N(0, 1)

distribution. Standardization:

Z = \frac{X - \mu}{\sigma}

Standard Normal Table, N(0,1) gives the probability that the standardized variable Z is less than or equal to $k_0$ , $P(Z \le k_0)$
Other cases
$P(Z \le -k_0) = 1 - P(Z \le k_0)$	$P(Z \ge k_0) = 1 - P(Z \le k_0)$
$P(Z \ge -k_0) = P(Z \le k_0)$	$P(k_0 \le Z \le k_1) = P(Z \le k_1) - P(Z \le k_0)$
$P(-k_0 \le Z \le -k_1) = P(Z \le k_0) - P(Z \le k_1)$	$P(-k_0 \le Z \le k_1) = P(Z \le k_1) - [1 - P(Z \le k_0)]$

Standard Normal Table, N(0,1) gives the probability that the standardized variable Z is less than or equal to $k_0$ , $P(Z \le k_0)$
Other cases
$P(Z \le -k_0) = 1 - P(Z \le k_0)$	$P(Z \ge k_0) = 1 - P(Z \le k_0)$
$P(Z \ge -k_0) = P(Z \le k_0)$	$P(k_0 \le Z \le k_1) = P(Z \le k_1) - P(Z \le k_0)$
$P(-k_0 \le Z \le -k_1) = P(Z \le k_0) - P(Z \le k_1)$	$P(-k_0 \le Z \le k_1) = P(Z \le k_1) - [1 - P(Z \le k_0)]$

The following table provides the area to the left of a value

z

, that is, the value of

P(Z \le z)

for a continuous random variable

Z

with a standard normal distribution

N(0, 1)

The following table provides the area to the left of a value

z

, that is, the value of

P(Z \le z)

for a continuous random variable

Z

with a standard normal distribution

N(0, 1)

Properties of the Normal distribution