Chapter 5: Continuous Random
Variables
Where We’ve Been


Using probability rules to find the
probability of discrete events.
Examined probability models for
discrete random variables.
McClave: Statistics, 11th ed. Chapter 5: Continuous
Random Variables
2
Where We’re Going



Develop the notion of a probability
distribution for a continuous random
variable.
Examine several important continuous
random variables and their probability
models.
Introduce the normal probability
distribution.
McClave: Statistics, 11th ed. Chapter 5: Continuous
Random Variables
3
5.1: Continuous Probability
Distributions


A continuous random variable can
assume any numerical value within
some interval or intervals.
The graph of the probability distribution
is a smooth curve called a



probability density function (pdf),
or
probability distribution.
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
4
5.1: Continuous Probability
Distributions

For continuous
random variables,
there are an infinite
number of possible
outcomes


P(X=a) = 0
Instead, find P(a<X<b)
 Table
 Software
 Integral calculus)
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
5
5.2: The Uniform Distribution



X can take on any
value between c and d.
For the rectangle to be
the pdf of uniform
distribution, the height
should be = 1/(d - c)
For two values a and b
ba
P ( a  x  b) 
d c
cabd
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
6
5.2: The Uniform Distribution
Mean:
cd

2
Standard Deviation:
d c

12
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
7
5.2: The Uniform Distribution
Suppose a random variable x is
distributed uniformly with
c = 5 and d = 25.
What is P(10 ≤ x ≤ 18)?
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
8
5.2: The Uniform Distribution
Suppose a random variable x is
distributed uniformly with
c = 5 and d = 25.
What is P(10 ≤ x ≤ 18)?
18  10
P(10  x  18) 
 0.40
25  5
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
9
5.3: The Normal Distribution

Closely approximates many situations


Perfectly symmetrical around its mean
The probability density function f(x):
1
f ( x) 
e
 2
[( x   ) /  ]2

2
µ = the mean of x
 = the standard deviation of x
 = 3.1416… and e = 2.71828 …
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
10
5.3: The Normal Distribution


Each combination of µ and  produces a
unique normal curve
The standard normal curve is used in
practice, based on the standard normal
random variable z (µ = 0, σ = 1), with the
probability distribution
f ( z) 
1
e
2
z2

2
The probabilities for z are given in Table IV
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
11
5.3: The Normal Distribution
P(0  z  1.00)  .3413
P(1.00  z  0)  .3413
P(1  z  1)
 .3413  .3413
 .6826
P(1  z  1.25) 
P(0  z  1.25)  P (0  z  1.00)
 .3944  .3413  .0531
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
12
5.3: The Normal Distribution
For a normally
distributed random
variable x, if we know
µ and ,
zi 
xi  

So any normally
distributed variable
can be analyzed
with this single
distribution
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
13
5.3: The Normal Distribution


Say a toy car goes an average of 3,000 meters
between recharges, with a standard deviation of 50
meters (i.e., µ = 3,000 and  = 50)
What is the probability that the car will go more than
3,100 meters without recharging?
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
14
5.3: The Normal Distribution


Say a toy car goes an average of 3,000 meters
between recharges, with a standard deviation of 50
meters (i.e., µ = 3,000 and  = 50)
What is the probability that the car will go more than
3,100 meters without recharging?
3100  3000 

P ( x  3100)  P z 

50


P ( z  2.00)  1  P ( z  2.00) 
1  .5  P (0  z  2.00) 
1  .5  .4772  .0228
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
15
5.3: The Normal Distribution

To find the probability for a normal random
variable …





Sketch the normal distribution
Indicate x’s mean
Convert the x variables into z values
Put both sets of values on the sketch, z below x
Use Table IV to find the desired probabilities
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
16
5.4: Descriptive Methods for
Assessing Normality

If the data are normal




A histogram or stem-and-leaf display will look like
the normal curve
The mean ± s, 2s and 3s will approximate the
empirical rule percentages
The ratio of the interquartile range to the standard
deviation will be about 1.3
A normal probability plot , a scatterplot with the
ranked data on one axis and the expected z-scores
from a standard normal distribution on the other
axis, will produce close to a straight line
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
17
5.4: Descriptive Methods for
Assessing Normality

IQR 22

 1.29
s
17
Errors per MLB team in 2003
Mean: 106
Standard Deviation: 17

IQR: 22



Frequency
Histogram
10
9
8
7
6
5
4
3
2
1
0
89  123
Frequency
89.8 102.6 115.4 128.2 More
Errors per team, 2003
22 out of 30: 73%
x  2 s  106  34
72  140
77

x  s  106  17
28 out of 30: 93%
x  3s  106  51
55  157
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
30 out of 30: 100%
18
5.4: Descriptive Methods for
Assessing Normality

3
Normal Quantile
2
1
0
A normal probability
plot is a scatterplot with
the ranked data on one
axis and the expected zscores from a standard
normal distribution on
the other axis
-1
-2
-3
60
80
100
120
140
160
Errors
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
19
5.5: Approximating a Binomial
Distribution with the Normal
Distribution


Discrete calculations may become very
cumbersome.
The normal distribution may be used to
approximate discrete distributions, such
as Binomial distribution.


The larger n is, and the closer p is to 0.5, the
better the approximation
Since we need a range, not a value, the
continuity correction must be used

A number r becomes r+0.5 or r-0.5
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
20
5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Calculate the mean plus/minus 3 standard deviations
  3  np  npq
If this interval is in the range 0 to n,
the approximation will be reasonably close
Express the binomial probability as a range of values
P( x  a)
P ( x  b)  P ( x  a )
Find the z-values for each binomial value
z
(a  0.5)  

Use the standard normal distribution to find
the probability for the range of values you calculated
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
21
5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Flip a coin 100 times and compare the binomial
and normal results
100  50 50
Binomial: P ( X  50)  
 0.5 0.5  0.0796
 50 
Normal:   100  0.5  50
  100  0.5  0.5  5
50.5  50 
 49.5  50
P(49.5  X  50.5)  P 
Z

5
5


P(0.10  Z  0.10)  0.0796
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
22
5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Flip a weighted coin [P(H)=0.4] 10 times and
compare the results
Binomial:
Normal:
10  5 5
P( X  5)    0.4 0.6  0.1204
5
  10  0.4  4
  10  0.4  0.6  1.55
5.5  4 
 4.5  4
P(4.5  X  5.5)  P 
Z

1.55 
 1.55
P(0.32  Z  0.32)  0.1255
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
23
5.5: Approximating a Binomial
Distribution with the Normal
Distribution
Flip a weighted coin [P(H)=0.4] 10 times and
compare the results
Binomial:
Normal:
10  5 5
P( X  5)    0.4 0.6  0.1204
5
The more p differs from 0.5,
  10  0.4  4
and the smaller n is,
  10  0.4  0.6  1.55
the less precise the
approximation will be
5.5  4 
 4.5  4
P(4.5  X  5.5)  P 
Z

1.55 
 1.55
P(0.32  Z  0.32)  0.1255
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
24
5.5: Approximating a Binomial
Distribution with the Normal
Distribution





The general rule of thumb for normal
approximation to binomial to be
appropriate is that
(1) n>20
and
(2) np and n(1-p) both >5
should hold.
McClave: Statistics, 11th ed. Chapter 5:
Continuous Random Variables
25