Normal Distribution
A lot of our real life experience could be
explained by normal distribution.
 Normal distribution is important for
statisticians for some other reasons.
 The mean and variance/standard
deviation make more sense.
 Different ways of calculating probability.

Normal Distribution

PDF:

1
f ( x) 
e
 2
( x   )2
2 2

= mean
  = standard deviation
  =3.14159
 e =2.71828
Normal distribution
The parameters of normal distribution:
 and 

 Shape: pdf and cdf.

Normal distribution

What can we say about the shape of the
bell curve?
◦
◦
◦
◦
1.
2.
3.
4.
Bell shaped.
Symmetric about the mean.
The highest point is the mean.
The tails are thin.
Normal distribution

What the parameters mean to us?
◦ Mean: Location of the center of the bell curve.
 If mean increases, the curve shifts to the right.
 If mean decreases, the curve shifts to the left.
◦ Standard deviation: Shape of the bell curve
(flat?, wide?, tall?)
Normal Distribution
What else can standard deviation tell us?
 Actually, the standard deviation of normal
distribution can tell us a lot more than the
standard deviations of other distributions.

Normal distribution

If we have a random variable that follows
a normal distribution, then:
◦ 68.3% of its values fall within ONE standard
deviation.
◦ 95.4% of its values fall within TWO standard
deviations.
◦ 99.7% of its values fall within THREE standard
deviations.
Normal distribution

Given a random variable, X, with mean 
and standard deviation  , we can create
a standardized version of this random
variable:
x
◦

◦ If we do this to a normally distributed random
variable, we get a standard normal random
variable.
◦ Usually, we use the letter Z to represent it.
Standard Normal Distribution
The standardized normal random variable,
Z, follows a standard normal distribution.
 Regardless of the original mean and
variance for X, the one before
standardization, the mean and standard
deviation of Z are:

and
 0
 1

Standard Normal Distribution

The pdf for Z is:

Therefore, whenever we talk about
standard normal distribution, we know
both its functional form and its
parameters.
f (Z ) 
1
e
2

z2
2
Standard Normal Distribution

For any random variable, we need to
know
◦
◦
◦
◦
1.
2.
3.
4.
How
How
How
How
is it defined?
can it be used?
to find probability under it?
to find its mean and variance?
Think about how to find probability under
normal distribution.
What is the use of standard
normal?

1. If we use the formula for mean and
variance:
   xf ( x)dx

   ( x   ) f ( x)dx
2
2


That is hard!!!
Standard normal distribution
2. Another way of finding probability
under normal distribution, using CDF.
 NOT CDF of any normal random variable,
but CDF of a standard normal random
variable.
 This function is usually denoted as:
 And that is defined as  ( Z )

( Z 0 )  P( Z  Z 0 )
Standard Normal Distribution







Some probabilities under standard normal
distribution:
P(Z<1)= ?
P(Z<0)=?
P(Z>0)=?
P(-1<Z<1)=?
P(Z<-2)=?
P(Z>3)=?
Standard Normal Distribution
 Be
prepared to draw
plots!
A tip to finding probability under
normal distribution
P(0<Z<0.5)=?
 P(1.5<Z<2.5)=?
 Probabilities like that can be found using
standard normal probability table, or Z
table. Appendix B in your textbook.

Some other probabilities under
standard normal distribution
If we have a normal random variable, X,
instead of standard normal random
variable to start with, we can always
standardize X to Z and look up the
probability in the Z table.
 Example: X~N(5,4), find the probability
P(X<13), P(X<8.6) and P(X<-3.5)
 Also, find P(2<X<9), P(X>10|X>3),
P(X>10|X>5), P(X<8|X<3) and
P(X<3|X<8)

How about just a normal r.v.?
A STAT301 midterm has a mean of 71 and
standard deviation of 4.8.
 1. What is the probability that someone’s
grade is between 60 and 80?
 2. What is the probability that someone’s
grade is greater than 72?
 3. What is the probability that someone’s
grade is below 50?

Example

If half of the class is going to get A in the
above course, so what will be the cutoff for
A?

This problem uses something we learned
before, the percentile.
This kind of problem can be solved using a
different form of the standardization formula:

In the formula, both  and 
and the percentile is given by Z.
X    Z
Example
are given

If 10% of students will get A, what is the
cutoff?

If 25% of students will get A, what is the
cutoff?
Example