Be able to calculate probabilities based on the Normal Distribution.
Explain why the Normal Distribution is so important.
3 The Probability Density Function 4 Notation
If a variable X is normally distributed with mean and standard deviation we write
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)
5 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
To help us we find the Cumulative Density Function F(A) P(XltA)
6 The Standard Normal Cumulative Distribution 7 Comparison of pdf to cdf 8 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
9 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.
10 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).
11 How do we actually solve problems
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)
12 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.
Calculate probabilities based on the Normal Distribution.
Explain why the Normal Distribution is important.
4.61a 4.62 a d 4.65 a e 4.69 4.72
Exam 1A 27 through 33
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