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The Normal Distribution

- MSIT3000
- Lecture 08

Objectives

- Be able to calculate probabilities based on the

Normal Distribution. - Explain why the Normal Distribution is so

important. - Text 4.7

The Probability Density Function

Notation

- If a variable X is normally distributed with mean

? and standard deviation ?, we write - X?N(?,?)
- If X has a mean of zero and a standard deviation

of one, we call X standard normally distributed

(and we usually use Z instead of X) - Z?N(0,1)

Calculating probabilities

- Recall that P(XA)0 for any specific value of A.

- This is true for all continuous probability

distributions. - Therefore we can only calculate probabilities of

intervals - P(XltA)
- To help us, we find the Cumulative Density

Function F(A) P(XltA)

The Standard Normal Cumulative Distribution

Comparison of pdf to cdf

Calculating probability

- To find the probability that X is less than A,

look up A in the CDF and read off the

probability. - We use a table to find a numerical value (see the

back of the text). - Problem the CDF is different for every

difference in ? or ?. - Solution convert to standard normal
- i.e. x to z

Converting X to Z

- If X is normally distributed, then
- is standard normally distributed.
- Therefore, P(XltA)P(Zlt(A-?)/?) is something we

can easily look up in the table.

Tricks for finding probabilities

- The normal distribution is symmetrical, so if ?

0 then P(X lt A) P( X gt -A ). - This is useful because the table does not have

entries for negative values of A. - P(X gt A)1 - P(X lt A)
- This is useful because the table only gives

probabilities for events such as (XltA), not

(XgtA).

How do we actually solve problems?

- Draw.
- Indicate the area you wish to find.
- Convert the probability you are interested in to

z-scores. - This is because every normally distributed random

variable transforms to the standard normal, so

only the standard normal is in your book. - Manipulate (use the tricks)
- P(altzltb)P(zltb)-P(zlta)
- P(zgta)P(zlt-a)

Why is the Normal Distribution so important?

- Almost every distribution you examine will be

closely related to the normal distribution. - Consider the Empirical Rule.
- Almost every random variable can be transformed

into another that is approximately normal. - It is (relatively) easy to calculate

probabilities for normally distributed random

variables by using the Standard Normal

distribution.

Conclusion

- Objectives addressed
- Calculate probabilities based on the Normal

Distribution. - Explain why the Normal Distribution is important.
- Problems
- 4.61a, 4.62 a d, 4.65 a e, 4.69, 4.72
- Exam 1A 27 through 33

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