Standardizing a normal distribution

Fortunately, no new tables are required for probability calculations regarding the general normal distributions. Any normal distribution can be converted to the standard normal by the following relation:

Rule:

Let X be a normally distributed random variable with mean and variance . Then random variable

has a standard normal distribution.

It follows that if a and b are any numbers with a < b, then

,

where Z is the standard normal random variable and denotes its cumulative distribution function.

Example:

If find the probability that X is greater than 16.

Solution:

The probability that X assumes a value greater than 16 is given by the area under the area under the normal curve to the right of x =16, as shown in Figure 4.19.

For x =16; .

The required probability is given by the area to the right of z =2.00.

.

Example:

Let X be a continuous random variable that has a normal distribution with and . Find the probability .

Solution:

The probability is given by the area from x =30 to x =39 under the normal curve.(Fig.4.20)

The z -values corresponding to x =30 and x =39 are

For x =30; ;

For x = 39; .

We calculate:

.

Example:

The number of calories in a salad on the lunch menu is normally distributed with mean and standard deviation . Find the probability that the salad you select will contain

a) more than 208 calories;

b) between 190 and 200 calories.

Solution:

Letting X denote the number of calories in the salad, we have the standardized variable

a)

.

b)

.

Дата добавления: 2015-08-05