ST 361 Normal Distribution



ST361: Ch1.4 Distribution of Continuous R.V.: Normal Distribution

Topics:

§1.4 Normal Distribution, and its density function, mean, variance

Standard Normal Distribution: (a) Calculating Probability

(b) Calculating Percentile

General Normal Distribution: (a) Calculating Probability

(b) Calculating Percentile

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I. Normal random variable/Normal Distribution

• A distribution for describing ______________ random variables

1. Normal density plot

[pic]

➢ Shape: ______________________________, centering at ______________; also median ____ mean

➢ Can be fully specified via two parameters: _______________ and _______________. The distribution is denoted by ________________

Ex.

[pic]

2. Normal density function (for your reference):

[pic]

Ex. If X ~ N (0,1) then f(x) is

II. Standard Normal Distribution

• Normal distribution with ________________________. Denoted by ______________

• Usually use _____ to denote a standard normal r.v.

• Why learn the standard normal distribution?

o Area under the normal curve can only be calculated numerically.

So statisticians have established a table that shows the left tail area under the standard normal curve of any given number (see the very first page of the textbook).

o Later we can use such table to solve for all normal distribution.

• Use the area table of standard normal curve

(1) Calculate probability

Ex. A variable Z ~ N(0, 1). Calculating the following probabilities:

1. P(Z [pic]1.25) =

2. P(Z [pic]-1.25) =

3. P(Z [pic]-1.25) =

4. P(Z [pic]2) =

► In general, _______________________________________

5. P(-.38 [pic] Z [pic].25) =

In general, ______________________________________________________

6. P(Z [pic]-6) =

(2) Obtain extreme values

Ex1. A variable Z ~ N(0, 1). Find the following z* that fulfills the probability:

1. P(Z[pic]z*) = 0.1

2. P(Z[pic]z*) = 0.5

3. P(Z[pic]z* or Z[pic]-z*) = 0.1

Ex2. Consider a standard Normal r.v. Z~N(0,1). At what value of z*, the area to the right is 2.5%?

Ex3. Consider a standard Normal r.v . Z~N(0,1). At what value of z*, the area between –z* and z* is 68%?

III. General Normal Distribution[pic][pic]

• If X has a normal distribution with mean [pic]and SD [pic], then we can standardize X to Z by

• Therefore,

[pic]

• Calculating probability and percentiles

Ex. A variable X ~ N(100, 5). Calculating the following probabilities:

1. P(90 [pic] X [pic]125) =

2. P( X[pic]98 ) =

3. Find the x* such that P( X [pic]x* )=0.1

► In general, _______________________________________

4. Find the range that contains the MIDDLE 90% of the observations

Ex. X is the diameter of tires, normally distributed with mean 575 and SD 5.

1. P(575 < X < 579)=

2. P(575 [pic]X [pic] 579)=

3. Find the diameter x* such that there are only 1% tires longer than this diameter

4. Find the tires that have most extreme 5% diameters.

IV. Putting everything together…. An overall example:

The diameter of a tire follows normally distribution with mean 575 and SD 5. We have 4 tires, and the diameters of these tires are independent of each other.

a) What is the probability that a tire has its diameter between 570 and 580?

b) What is the probability that all 4 tires have diameters between 570 and 580?

c) What is the probability that at least one tire is not between 570 and 580?

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A

B

C

D

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