# Normal Distribution

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A continuous random variable X is normally distributed with parameters and denoted if and only if its **density** function is:
the distribution function is :
The Normal distribution depends on two parameters and , which are the expected value and the standard deviation of the random variable X.
**Expected value**, **variance** and **standard deviation**:
Two important **properties** of Normal random variables:

Linear transformation

Let X be Normally distributed, and Y be a linear combination of X: . Then, the random variable Y has also Normal distribution:

Y N(a + b, b )

The values of the parameters of the transformed random variable follow from the rules for calculating with expecting values and variances:

E(a + bX) = a + b E(X)

Var(a + bX) = Var(X) = .

Reproduction property

Let us consider n random variables with Normal distributions:

The sum of independent, normally distributed random variables , i.e.

for at least one i, is again normally distributed.

The following diagrams displays a density and distribution function for a N(2;1) random variable. Density:

The distribution function of N(2;1):

**Standardized random variable:**
The random variable Z denotes a standardized random variable, which has been centred at its mean and scaled by its standard deviation.If X is normally distributed, then Z also has a Normal distribution.
**Standardized Normal distribution:**
The distribution of Z is usually denoted as standardized Normal distribution N(0;1).
The density function of a standardized Normal distribution:
The distribution function of a standardized Normal distribution:
Expected value and variance of standardized Normal distribution:
E(Z) = 0 Var(Z) = 1
The density and distribution function for a standardized normal random variable are plotted in the following figures.
Density of N(0;1)

Distribution function of N(0;1)

The relation between the distribution N() and the standardized Normal distribution:
which implies:
**Confidence interval:**
A confidence interval for the random variable X is the interval with boundaries and , which will contain the value of the random variable X with probability 1 - , i.e. (1 -) 100% of all values of X will fall in this interval and 100% will fall outside this interval. 1- is usually referred to as the confidence level.For known values of the expected value of X, the interval is constructed to make the probability that X falls outside this region (there are 2 such regions) with probability /2. We call the interval
[] = []
the (symmetric) confidence interval with confidence level
P() = 1 - .
To stress the importance of the standard deviation, as the parameter of scale, the deviation of X from its expected value is often measured in multiples of . The confidence interval has then this form:
[ - c X + c]
If the random variable X is N(), then for x = + c the following holds:
and P(Z z) = (z) = 1 - /2 .
The critical value for the probability 1 - /2 can be obtained from the tabulated values of a standardized Normal distribution.
Using these values, we can obtain the confidence interval for a normally distributed random variable:
[ X ]
and the probability of “this interval”:
P( X ) = 1 -
The confidence interval for normally distributed random variable:

We have P(-z Z z) = P(Z z) - P(Z -z) = P(Z z) - [1 - P(Z z)] = 2P(Z z) -1 , which implies that . For given z we can calculate the confidence levels of the interval:

for | |

for | |

for |

On the other hand, we could also find the value z that produces the desired confidence level 1-, e.g. = 0.95, z = 1.96. The Normal distribution is described by two parameters which imply its:

- shape
- location and
- scale (variance)

In this interactive example, you can choose different values of these parameters and observe their effect on the density function of a normal random variable. We recommend that you only change one parameter at time to better observe their effects on the distribution function. The density function of a standard Normal distribution is presented (in black) to provide a further reference point. In addition, you can also calculate the probability that falls in some interval. Let us consider random variable with Normal distribution .

We want to compute for :

There is a 99.38% probability that the random variable is smaller than 125.

We want to calculate the probability for :

There is a 5.94% probability that the random variable is greater than 115.6.

Let us calculate the probability for :

The random variable is smaller than 80 with probability of 2.275% .

Let us compute for :

The probability that the random variable is greater than 94.8 is 69.85% .

We compute the probability for and :

The random variable falls in the interval with probability 86.8% .

Let us calculate for and (centered probability interval):

The random variable falls into the interval with probability 95% .

We want to calculate an interval, which is symmetric around the expected value, such that it will contain 99% of the realizations of :

For the value (the probability) 0.995 we find in the tables of the distribution function of standard Normal distribution function that .. This implies: take .

The random variable falls into the interval with a 99% probability .

Let us find an such that 76.11% of the realizations of are smaller than :

For the value 0.7611 we obtain from the standard Normal distribution tables that . Hence: so that .

There is a 76.11% probability that the random variable will be smaller than 107.1.

We calculate such that 3.6% of realizations of is greater than :

Since , using the standard Normal distribution tables the value for the probability 0.964. Hence, so that .

There is a 3.6% probability that the random variable is greater than 118.

*The Normal distribution is one of the most important continuous distributions because:*

*approximate normality can be assumed in many applications**it can be used to approximate other distributions**many variables have normal distributions if there is a large number of observations*

A random variable with a Normal distribution can take all values between - and + The Normal distribution is also sometimes referred to as a Gaussian distribution. The density of Normal distribution is sometimes called the Bell curve.The formulas for the density (or the distribution function) imply that a Normal distribution will depend on the parameters .. By varying these parameters we can obtain a range of distributions. The following diagram shows 5 normal densities with various parameters .

The parameter specifies the location of the distribution. If we change the parameter , the location of the distribution will shift but its shape remains the same.By increasing or decreasing the parameter , the density ”spreads” or ”concentrates”. Large values of , produce flatter and wider densities. Small values of produce distributions that are narrow and tight.
Other **properties of the Normal distribution :**

- the density has global maximum (the mode) at point
- the density is symmetric around the point . The symmetry implies that the median is .
- the density has inflexion points at and
- the density is asymptotically equal to 0 as or .

The digram contains a plot of a distribution

**Standard Normal distribution:**
Tabulating the distribution function of the Normal distribution for all values of and is not possible.However, since we can transform a Normal random variable to obtain another Normal random variable we need only tabulate one distribution. The obvious choice is the Normal distribution with expected value , and standard deviation , .
This distribution is called a standard Normal distribution, denoted – distribution. The corresponding random variables are usually denoted by the letter .The random variable is the random variable centered at its mean and divided by its standard deviation. Hence and . If is normally distributed, then also has a (standard) Normal distribution.
The standard Normal distribution is important because each random variable with arbitrary Normal distribution can be linearly transformed to a random variable with standard Normal distribution.In most tables for the density and distribution function of the standard Normal distribution, you can find only positive values of . The tables of standard Normal distribution for negative is unnecessary since the Normal distribution is symmetric.