### 4.4    The Normal Distribution

Def: Let X be a continuous random variable.  If its density is

then X has the normal distribution w/ parameters m, s.
• called a normal random variable
• the parameters  m  and  s  are the mean and standard deviation (hard to show directly; use moment generating function (below))
• graph: "bell-shaped curve", w/ maximum at  x = m,  inflection points at  x = m ± s  (use calculus to show (prob. #43)):

•
• it is true that ,    but requires some multivariable calculus tricks to show this!

Moment generating function:

(see text for derivation)

Can use this to find mean, variance:

Can similarly find the second moment using the second derivative; get

E(X2)  =  m2 + s2
giving   s2  =  s2.
Thus the parameters m, s in the density function are in fact the mean and s.d.!

ex:

IQ scores are assigned in such a way that they are normally distributed with mean 100 and standard deviation 15. Let  X  be the IQ score of a person selected at random. What's the density function of X?

Since  m = 100  and  s = 15,  we get

Graph:

Note:  there are lots of normal distributions, one for each value of  m, s
•   determines where the center of the distribution will be
• the larger  s  is, the broader the distribution

Def:  The normal distribution with  m = 0, s = 1 is called standard normal distribution; use variable Z to denote it.
Density function for  Z  is

f(x)   =

Finding probabilities for normal distributions

ex:

Consider IQ scores, as above.
What's the probability that a randomly selected person will have an IQ score less than or equal to 100?
Easy to find using symmetry:
P(X <= 100)  =  area under the density curve to the left of 100  =  1/2

What's the probability that a randomly selected person will have an IQ score between 110 and 130?
P(110 <= X <= 130)   =   area under density curve from 110 to 130

This is hard to find! We can't find an antiderivative to use to evaluate the integral.

• Could use numerical methods (such as the trapezoid rule or Simpson's rule) to find the value.
• Instead, use tables for the standard normal distribution

Approach:
• tabulate values for the standard normal dist; see table V, appendix A, p. 637;

• the table gives values of  P(Z <= z)  for various values of z
• use the standardization theorem to transform a question about a non-standard normal random variable  X into one about the standard normal random variable  Z:

• Theorem   If X is normal, with mean m,  s.d. s,  then  is a standard normal random variable.  (Proof: find the moment generating function of   from the moment generating function of X using our rules for moment generating functions discussed earlier, and see that it is the moment generating function of a standard normal random variable.)

ex:

Consider IQ scores, as above.

What's the probability that a randomly selected person will have an IQ score of 80 or lower?

Want  P(X <= 80).
Let
Then
when X = 80,    =   -1.33,
so
P(X <= 80)  =  P(Z <= -1.33)  =  .0918  from the table.
Thus about 9% of people have IQ scores less than 80.

What's the probability that a randomly selected person will have an IQ score between 110 and 130?
Want  P(110 <= X <= 130);
again, let
Then
when X = 110,    = .67,
when X = 130,    = 2.00,
so
P(110 <= X <= 130)  =  P(.67 <= Z <= 2.00)
=   P(Z<= 2.00)  -  P(Z < .67)
(i.e., the area to the left of 2.00 minus the area to the left of .67)
=  .9772  -  .7486   from the table
=  .2286 .
So about 23% of people have IQ scores between 110 and 130.

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