4.3. The normal distribution

The normal distribution is one of the many probability distributions that a continuous random variable can possess. The normal distribution is the most important and most widely used of all the probability distributions. A large number of phenomena in the real world are normally distributed either exactly or approximately.

The probability density function for a normally distributed random variable X is:

for

where and are any number such that and , e=2.71828…. and are constants.

A normal probability distribution, when plotted, gives a bell-shaped curve such that

1. The total area under the curve is 1.0.

2. The curve is symmetric about the mean.

3. The two tails of the curve extend indefinitely.

1. The total area under a normal curve is 1.0 or 100%, as shown

in Figure 4.6.

2. A normal curve is symmetric about the mean, as shown in Figure 4.7. Consequently 0.5 of the total area under a normal curve lies on the left side of the mean and 0.5 lies on the right side of the mean.

3. The tails of a normal distribution curve extend indefinitely in both

directions without touching or crossing the horizontal axis.

Although a normal curve never meets the horizontal axis, beyond the points

represented by and it becomes so close to this axis that the

area under the curve beyond these points in both directions can be taken as

virtually zero. These areas are shown in Figure 4.8.

Remark:

There is not just one normal distribution curve but rather a family

of normal distribution curves. Each different set of values of and gives

a different normal distribution.

The value of determines the center of a normal distribution on the

horizontal axis. The three distribution curves drawn in Figure 4.9 have the

same mean but different standard deviations.

The value of gives the spread of the normal distribution curve. The three

normal distribution curves in Figure 4.10 have different means but the same

s tandard deviation.

Properties of the normal distribution:

Suppose that the random variable X follows a normal distribution. Then the

following properties hold:

1. The mean of the random variable is :

2. The variance of the random variable is

3. By knowing the mean and standard deviation (or variance) we can define

the normal distribution by using the notation:

4.3.1. Cumulative distribution function of the

normal distribution

Suppose that X is a normal random variable with mean , and variance ,

t hat is .

Then the cumulative distribution function

This is the area under the normal

probability density function to the left

of as illustrated in Figure 4.11.

As for any proper density function,

the total area under the curve is 1,

that is

and .

Let a and b be two possible values of X, with a<b.

Then

The probability is the area under corresponding density function between

a and b as shown in Figure 4.12.