**Exhibit 34.17 **Normal distribution curve.

The normal distribution (Exhibit 34.17) is said to represent an elementary
“truth about the general nature of reality”. It is a family of distributions, each defined by two
parameters — mean (μ) and standard deviation (s).

The probability distribution function of a random variable *X∼N(μ,σ)* is:

$$f(x)=\frac{1}{\sqrt{2πσ^2}} e^{-\frac{(x-μ)^2}{2σ^2}}$$

### Standard Normal Distribution

The normal distribution *X∼N(0,1)*, *μ = 0* and *σ = 1*, is called
the standard normal distribution, and its distribution function is:

$$f(x)=\frac{e^{-x^2/2}}{\sqrt{2π}}$$

A value from any normal distribution can be transformed into its corresponding value
on a standard normal distribution using the following formula:

$$Z =\frac{X - μ}{σ}$$

where Z is the value on the standard normal distribution, X is the value on the
original distribution, μ is the mean of the original distribution, and σ is the standard deviation
of the original distribution.

$$E(Z)= E\biggl(\frac{X - μ}{σ}\biggr)=\frac{E(X) - μ}{σ}=0$$
$$Var(Z)=Var\biggl(\frac{X - μ}{σ}\biggr)=\frac{Var(X)}{σ^2}=1$$

To computing probabilities for normal distributions, convert to standard value and
lookup the distribution function tables for the standard normal distribution:

$$P(X≤b)→P\biggl(Z≤\frac{b - μ}{σ}\biggr)$$

*Note: The probabilities may also be obtained using the NORM.DIST function in Excel.
The Excel NORM.INV function returns the variable value, given the parameters and the probability.*

**Exhibit 34.18 **90% of the observations fall within a range of ±1.65 standard deviation
from the mean.

If a random variable follows a normal distribution, we can expect that:

- 68% of the variable’s observations fall within a range of ±1 standard deviation
from the mean.
- 90% of the observations fall within a range of ± 1.65 standard deviations from
the mean, as depicted in Exhibit 34.18.
- 95% of observations lie within 1.96 standard deviations from the mean.

### Approximating Binomial with Normal

According to the Central Limit Theorem, the normal
distribution can be used as an approximation to the binomial distribution, if the sample size is
large enough.

If *X* is Binomial *(n, p)*:

$$E(X) = np$$
$$Var(V) = np(1-p)$$

The approximate equivalent normal distribution is *N[np,√(np(1-p))], provided
np≥ 5 and n(1- p)≥ 5* (thumb rule).