Transcript Chapter 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 
2
Population Variance =  2   x    f x    x 2 f x    2
Population Standard Deviation =    2
8.2 The Uniform Distribution
The density function for the uniform distribution is as
follows:
1
f x  
a

x

b
for
ba
Calculating descriptive statistics
ab
Population Mean =  
2
Population variance =
2


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
Example: Travel time from Lexington KY to Columbus OH
is a uniform distributed between 200 and 240 minutes
 Give the density function
 Find the mean.
 Find the median.
 Find the variance.
 Find the standard deviation.
 Find the probability of arriving in less than 225 minutes.
8.3 The Normal Distribution
Density function for the normal distribution is as follows:
f x  
1
2 2
  x   2
e
2 
2
The normal distribution is a 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
measures the spread of the bell. The larger
is the
more spread out the bell.


For the normal the mean, median, and mode are equal.
The value of  and  characterize which normal
distribution we are using
The normal distribution with   0 and   1 is called
the standard normal distribution. (This is used to
calculate probabilities for all normal distributions)
If X is normal with mean  and standard deviation
then
Z
x

is standard normal.
,
Examples
Draw Pictures of desired areas when doing problems!!!
P(0  Z  1.55) 
P(0  Z  1.96) 
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
Examples
PZ  1.64 
P(Z  1.64) 
P(Z  1.64) 
P(2.32  Z  0) 
P(2  Z  2) 
P(1.41  Z  2.18) 
Notice that Probabilities in the table stop at 3.9 with an
area of .5000. Beyond this Z value you will always have
close to .5 the area.
8.4 Calculating Areas Under Any Normal
Curve
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)
Find P ( X  130)
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  11.83)