Saturday, January 26, 2008

Lecture 4 - Ch 6 - The Standard Normal Distribution

Standardizing the Normal Distribution

As you can see from the formula for the Normal Distribution in the previous post, the normal distribution has two parameters - the mean (mu) and the standard deviation (sigma). Other than those parameters, it's just a (complicated) function of x.

It should be pretty clear that if we make the mean = 0 and standard deviation = 1, the formula would become much simpler. We call it the standard normal distribution and it looks like this:Much simpler, eh? Well, it's not too bad. But it still requires a scientific calculator to compute. The weird thing about the standard normal distribution is that it does not describe the probability of any one value, but rather the probability of the result of an experiment falling between any two values is defined by the area under the curve of the standard normal distribution.

You know what that means? We have to integrate it! OOOOhh boy!

Well, fortunately for us, some math majors have taken it upon themselves to integrate this formula for us and calculate the definite integral between 0 and almost any positive value. That means we can use these handy-dandy tables to find out the probability of x falling between 0 and any value.

Finding the probability of x falling between a and b

Looking up Pr(0 < x < a ) in the table is helpful, but often we want to know the probability of x falling between two values, a and b, or Pr (x > a) or Pr(x < a). We can easily use the values in the table to determine any of these probabilities.

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