Download Ch 6 The Normal Distribution and Sampling Distributions

Ch 8 Some Useful Continuous Probability Distributions
8.1 Properties of Continuous Probability Distributions
A smooth curve known as the density function, f x  is used to represent the probability
distribution of a continuous random variable.
The curve must never fall below the x-axis … f x   0 for all x
The total area under the curve must be 1 …
 f x   1
For continuous random variables we assign probability to intervals. (Not Points)
P(a  x  b)  the area under the curve between a and b.
With continuous variables, each point has probability zero
P( x  a)  0
P ( x  b)  0
Thus for continuous variables
P ( a  x  b)  P ( a  x  b)
For continuous distributions
Population Mean =    xf  x 
Population Variance =  2   x    f  x    x 2 f x    2
2
Population Standard Deviation    2
8.2 The Uniform Distribution
The density function for the uniform distribution is as follows
1
f x  
for a  x  b
ba
Calculating descriptive statistics
ab
Population Mean =  
2
b
   xf  x 
a
b
1
x
ba
a


b
1
x2
2b  a  a

1
1
b2 
a2
2b  a 
2b  a 
b2  a2

2b  a 
b  a b  a 

2b  a 
ab

2
Population Median = 
Population variance =  2 
b  a 2
12
b
   x 2 f x    2
2
a
b
1
x2   2
ba
a


b
1
x3   2
3b  a  a

1
1
b3 
a3   2
3b  a 
3b  a 

b3  a3  a  b 


3b  a   2 

2
b  a b 2  ab  a 2   a 2  2ab  b 2
3b  a 
4
b 2  ab  a 2 a 2  2ab  b 2

3
4
2
2
2
4b  ab  a   3a  2ab  b 2 

12
2
2
b  2ab  a

12
b  a 2

12

Population standard deviation =    2
The probability that a value is between c and d is
d c
Pc  x  d  
where a  c  d  b
ba
d
 f x 
c
d
1
ba
c


d
1
x
ba c
d
c

ba ba
d c

ba

Example
The travel time from Lexington KY to Columbus Ohio is uniformly distributed between
200 and 240 minutes.
Find the mean.
a  b 200  240 440



 220
2
2
2
Find the median.
 220
Find the variance.
2
2
2



b  a
240  200
40
1600
2
 



 133.333
12
12
12
12
Find the standard deviation.
   2  133.333  11.547
Find the probability of arriving in less than 225 minutes.
d  c 225  200 25
P200  x  225 


 .625
b  a 240  200 40
8.3 The Normal Distribution
The density function for the normal distribution is as follows:
f x  
  x   2
1
2
2
e
2 
2
The normal distribution is a very common type of continuous distribution.
 It is a bell shaped curve. The bell is symmetric about the mean of the random
variable  .
 The standard deviation of the random variable  affects the spread of the bell. The
larger  is the more spread out the bell.
 The mean, median, and mode are all equal for the normal distribution.
The value of
 and  characterize which normal distribution we are talking about.
The normal distribution with   0 and   1 is called the standard normal distribution.
(This is used to calculate normal probabilities)
Fact: If X is normal with mean  and standard deviation  , then Z 
x

is standard
normal.
Explain why.
Areas under the Standard Normal Curve
Look at Standard Normal Distribution table in the back cover of the book; table gives
P(0  Z  a ) for the standard normal distribution
Draw graph and show area
Examples
Draw Pictures of desired areas when doing problems!!!
P(0  Z  1.55)  .4394
P(0  Z  1.96)  .4750
Facts:
Total area under the curve is 1
Curve is symmetric about 0
P( Z  0)  P( Z  0)  1
2
Combining these facts with the table allows us to compute all probability statements for Z
Example
PZ  1.64  .5  P0  Z  1.64  .5  .4945  .9495
P( Z  1.64)  .5  P(0  Z  1.64)  .5  .4495  .0505
P( Z  1.64)  P( Z  1.64)  .0505 by symmetry
P(2.32  Z  0)  P(0  Z  2.32)  .4898
P(2  Z  2)  2  P(0  Z  2)  2  .4772  .9544
Does this make sense by the Empirical Rule?
P(1.41  Z  2.18)  P(0  Z  2.18)  P(0  Z  1.41)  .4854  .4207  .0647
Notice that Probabilities in the table stop at 3.9 are .5000. Beyond this Z value you will
always have close to .5 the area.
8.4 Calculating Areas Under Any Normal Curve
We have learned how to compute probabilities for the standard normal. We will now
compute probabilities for any normal.
Fact: If x is normal with mean  and standard deviation  , then Z 
x

is standard
normal.
Write probability statement for X
Rewrite in terms of Z
Example
The distribution of IQ scores for the general population is approximately normal with
  100 and   10 .
x = IQ score of randomly selected person
Find P(100  X  120)
Draw Picture
P (100  X  120)
100  100
120  100
 P(
X 
)
10
10
 P (0  Z  2)
 .4772
Find P ( X  130)
Draw Picture
P( X  130)
130  100
 P( Z 
)
10
 P( Z  3)
 .5  P(0  Z  3)
 .5  .4987
 .0013
very unlikely
Example
Suppose the amount of Pepsi in a “12 oz” can has a Normal distribution with   12 oz.
and   .1 oz.
x  amount of Pepsi in a Randomly selected can
Find P( X  11.90)
Draw Picture
P ( X  11.90)
11.9  12
 P( Z 
)
.1
 P ( Z  1)
 .5  P0  Z  1
 .5  .3413
 .1587