Chapter 4
Continuous Random Variables and Probability Distributions
 4.1 - Probability Density Functions
 4.2 - Cumulative Distribution Functions and
Expected Values
 4.3 - The Normal Distribution
 4.4 - The Exponential and Gamma Distributions
 4.5 - Other Continuous Distributions
 4.6 - Probability Plots
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
“Density”
Pop vals
pmf
x
p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
Total Area = 1
f ( x)
p ( x)
(height) (area)
p(x) = Probability that the
random variable X is equal
to a specific value x, i.e.,
|
x
p(x) = P(X = x)
“probability mass
function” (pmf)
x
(width)
p( x)  f ( x) x
|
x
X
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop vals
pmf
x
p(x)
x1
p(x1)
F(x1) = p(x1)
x2
p(x2)
F(x2) = p(x1) + p(x2)
x3
p(x3)
F(x3) = p(x1) + p(x2) + p(x3)
⋮
⋮
⋮
Total
1
increases from 0 to 1
cdf
F(x) = P(X  x)
“staircase graph”
Total Area = 1
F(x) = Probability that the
random variable X is
less than or equal to a
specific value x, i.e.,
F(x) = P(X  x)
“cumulative distribution
function” (cdf)
|
x
X
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop vals
pmf
cdf
x
p(x)
x1
p(x1)
F(x1) = p(x1)
x2
p(x2)
F(x2) = p(x1) + p(x2)
x3
p(x3)
F(x3) = p(x1) + p(x2) + p(x3)
⋮
⋮
⋮
Total
1
increases from 0 to 1
F(x) = P(X  x)
Calculating
“interval probabilities”…
F(b) = P(X  b)
F(a–) = P(X  a–)
F(b) – F(a–) =
P(X  b) – P(X  a–)
= P(a  X  b)
b
  p(x)
a
| |
a–a
|
b
X
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop vals
pmf
x
p(x)
x1
p(x1)
F(x1) = p(x1)
x2
p(x2)
F(x2) = p(x1) + p(x2)
x3
p(x3)
F(x3) = p(x1) + p(x2) + p(x3)
⋮
⋮
⋮
Total
1
increases from 0 to 1
Calculating
“interval probabilities”…

F(b) = P(X  b)
F(a–) = P(X  a–)
b
a
cdf
F(x) = P(X  x)
f ( x) dx  F (b)  F (a )
b

f
(
x
)

x

F
(
b
)

F
(
a
)

F(b) – F(a–) =
a
p( x)
P(X  b) – P(X  a–)
= P(a  X  b)
b
  p(x)
a
| |
a–a
|
b
X
FUNDAMENTAL
THEOREM OF
CALCULUS
(discrete form)
Reconsider the following discrete random variable…
Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”
X is uniformly distributed over 1, 2, 3, 4, 5, 6.
Probability Table Cumul Prob
Probability Histogram
P(X = x)
P(X  x)
x
p(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
P( X  4)  1 6
1
6
Reconsider the following discrete random variable…
Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”
X is uniformly distributed over 1, 2, 3, 4, 5, 6.
Probability Table Cumul Prob
Probability Histogram
P(X = x)
P(X  x)
x
p(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
“staircase graph”
from 0 to 1
P( X  4)  1 6
1
7
Reconsider
Consider
thethe
following
following
continuous
discrete random
randomvariable…
variable…
Example: X = “Ages
“value of
shown
on afrom
single
random
toss
of a fair
children
1 year
old to
6 years
old”die (1, 2, 3, 4, 5, 6)”
3, 4, 5,[1,
6. 6].
Further suppose that X is uniformly distributed over 1,
the2,interval
Probability Table Cumul Prob
Probability Histogram
P(X = x)
P(X  x)
x
p(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
P( X  4)  1 6
1
8
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
3, 4, 5,[1,
6. 6].
Further suppose that X is uniformly distributed over 1,
the2,interval
Probability Table Cumul Prob
Probability Histogram
P(X = x)
P(X  x)
x
f(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability a
ofchild
rolling
is a4 4?”
years old?”
P( X  4) 
 16
1
9
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Example:
X = “reaction time”
“Pain Threshold” Experiment:
Volunteers place one hand on metal
plate carrying low electrical current;
measure duration till hand withdrawn.
Time
Time
intervals
intervals
= 1.0
= 5.0
0.5
2.0
1.0
secs
secs
“In the limit…”
f ( x)
we obtain a
density curve
Total Area = 1
SAMPLE
In principle, as # individuals in samples
increase without bound, the class
interval widths can be made arbitrarily
small, i.e, the scale at which X is
measured can be made arbitrarily fine,
since it is continuous.
10
“In the limit…” we obtain a density curve
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f(x) = probability
density function (pdf)
• f(x)  0
• Area = 1
f ( x)
00 F(x) increases
continuously
from 0 to 1.
x
x
x
As with discrete variables, the density f(x) is the height, NOT the probability p(x) = P(X = x).
In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ???
(Later…)
However, we can define “interval probabilities” of the form P(a  X  b), using cdf F(x).
11
“In the limit…” we obtain a density curve
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
F(b)
f(x) = probability
density function (pdf)
F(b)  F(a)
F(a)
• f(x)  0
• Area = 1
f ( x)
a
b
F(x) increases
continuously
from 0 to 1.
a
b
As with discrete variables, the density f(x) is the height, NOT the probability p(x) = P(X = x).
In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ???
(Later…)
However, we can define “interval probabilities” of the form P(a  X  b), using cdf F(x).
12
“In the limit…” we obtain a density curve
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
F(b)
f(x) = probability
density function (pdf)
F(b)  F(a)
F(a)
• f(x)  0
• Area = 1
f ( x)
a
b
F(x) increases
continuously
from 0 to 1.
a
b
An “interval probability” P(a  X  b) can be calculated as the amount of area under
the curve f(x) between a and b, or the difference P(X  b)  P(X  a), i.e., F(b)  F(a).
(Ordinarily, finding the area under a general curve requires calculus techniques…
unless the “curve” is a straight line, for instance. Examples to follow…)
13
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20 > 0
Density
f ( x)

Total Area = 1
1 Check?
1
1
1
6 Base 6= 6 – 16= 5 6
Height = 0.2
1
6
1
6
5  0.2 = 1 
X
“What is the probability of
that
rolling
a random
a 4?” child is 4 years old?” doesn’t mean…..
P( X  4)
4.000000000......)
 16
A single value is one point out of an infinite
continuum of points on the real number line.
The probability that a continuous
random variable is exactly equal to
any single value is ZERO!
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20
Density
f ( x)
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of
rolling
a 4?” child is 4between
4 and
5 years
old?”
that
a random
years old?”
actually
means....
P(4
( XX4) 5) = (5 – 4)(0.2) = 0.2
NOTE: Since P(X = 5) = 0, no change for P(4  X  5), P(4 < X  5), or P(4 < X < 5).
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  0.20
Density
f ( x)
For any x, the area
under the curve is
1
6
1
6
1F(x) =10.2 (x1– 1). 1
6
6
6
6
X
x
x
or F ( x)   0.2 dt
1
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  0.20
F(x) = 0.2 (x – 1)
Density
f ( x)
For any x, the area
under the curve is
1
6
1
6
F(x) increases
continuously
from 0 to 1.
1F(x) =10.2 (x1– 1). 1
6
6
6
6
(compare with
“staircase graph”
for discrete case)
X
x
x
or F ( x)   0.2 dt
1
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  0.20
F(x) = 0.2 (x – 1)
Density
f ( x)
F(5) = 0.8
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of
rolling
a 4?” child is under 5 years old?
that
a random
F (5)  P( X  5)  0.2 (5  1)  0.8
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  0.20
F(x) = 0.2 (x – 1)
Density
f ( x)
1
6
1
6
1
6
1
6
1
6
1
6
F(4) = 0.6
X
“What is the probability of
rolling
a 4?” child is under 4 years old?
that
a random
F (4)  P( X  4)  0.2 (4  1)  0.6
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  0.20
F(x) = 0.2 (x – 1)
Density
f ( x)
F(5) = 0.8
1
6
1
6
1
6
1
6
1
6
1
6
F(4) = 0.6
X
“What is the probability of
rolling
a 4?” child is between 4 and 5 years old?”
that
a random
P(4  X  5)  P( X  5)  P( X  4)
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  0.20
F(x) = 0.2 (x – 1)
Density
f ( x)
F(5) = 0.8
1
6
1
6
1
6
1
6
1
6
0.2
1
6
F(4) = 0.6
X
“What is the probability of
rolling
a 4?” child is between 4 and 5 years old?”
that
a random
P(4  X  5)  P( X  5)  P( X  4)
= F(5)  F(4) = 0.8 – 0.6 = 0.2
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Density
f ( x)
f ( x)  .08 ( x  1)  0 
1
Base
Height
 1)  (0.4)
Area = (6
2
=1 
0.4
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Cumulative Distribution Function F(x)
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  .08 ( x  1)
Density
f ( x)
F ( x)
x
base
height
1
( x  1) .08( x  1)
2
 .04 ( x  1) 2
F ( x) 
i.e.,

x
1
.08(t  1) dt
F ( x)
x
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Cumulative Distribution Function F(x)
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  .08 ( x  1)
Density
f ( x)
F ( x)
base
height
1
( x  1) .08( x  1)
2
 .04 ( x  1) 2
F ( x) 
i.e.,

x
1
.08(t  1) dt
F (5)
F (4)
x
“What is the probability that a child is under 4 years old?”
“What is the probability that a child is under 5 years old?”
“What is the probability that a child is between 4 and 5?”
P( X  4)  F (4)
P( X  5)  F (5)
P(4  X  5) 
A continuous random variable X
Cumulative probability function (cdf)
In
summary…
x
corresponds to a probability density
F ( x)  P( X  x) 
f (t ) dt
function (pdf) f(x), whose graph is a

density curve. f(x) is NOT a pmf!
F ( x)  f ( x)

f ( x)  0
f ( x)



f ( x) dx  1
Fundamental
Theorem of
Calculus
P( X  any constant a)  0, not f (a)
F(x) increases
continuously
from 0 to 1.
b
P(a  X  b)   f ( x) dx  F (b)  F (a )
Moreover…
a
25
A continuous random variable X
Cumulative probability function (cdf)
In
summary…
x
corresponds to a probability density
F ( x)  P( X  x) 
f (t ) dt
function (pdf) f(x), whose graph is a

density curve. f(x) is NOT a pmf!
F ( x)  f ( x)

f ( x)
0
  E[ X ]   f ( x) dx
x f1( x) dx



  E ( X   )   
2
2

Fundamental
Theorem of
Calculus

( x   ) F(x)f increases
( x) dx
2

continuously
from 0 to 1.
 E  X   E[ X ]   x f ( x) dx  

2
2
P( X  any constant a)  0, not f (a)
2
b
2
P(a  X  b)   f ( x) dx  F (b)  F (a )
Moreover…
a
26
SECTION 4.3 IN POSTED LECTURE NOTES
28
Four Examples: 1
For any b > 0, consider the following probability density function (pdf)...
Determine the cumulative distribution function (cdf)
2
 x, 0  x  b
f ( x)   b 2
F ( x)  P( X  x)
0,
else
For any x < 0, it follows that F ( x)  P( X  x)  0.
2
For any 0  x  b, it follows that…

b
F ( x)  P( X  x)  x 2 b2
2
without calculus...
x
b2
1
 2
0  ( x  0)  2
2
b
2
 x
x  2
 b
with calculus...

|
x
0
b


f (t ) dt  0  
x
0
2
x2
t dt  2
2
b
b
29
Four Examples: 1

For any b > 0, consider the following probability density function (pdf)...
Determine the cumulative distribution function (cdf)
2
 x, 0  x  b
f ( x)   b 2
F ( x)  P( X  x)
0,
else
For any x < 0, it follows that F ( x)  P( X  x)  0.
2
For any 0  x  b, it follows that…

b
F ( x)  P( X  x)  x 2 b2
2
x
2
b
Note: F (0)  0 2 b 2  0
F (b)  b 2 b 2  1
For any x  b, it follows that…
F ( x)  P( X  x)  1  0  1.
0
|
x
b
30
Four Examples: 1
For any b > 0, consider the following probability density function (pdf)...
Determine the cumulative distribution function (cdf)
2
 x, 0  x  b
x0
f ( x)   b 2
F ( x)  P( X  x)  0,
 x 2
0,
else
 2 , 0 xb
2
b


xb
b
 1,
2
x
2
b
1 2
x
2
b
0
b
0
b
31
Four Examples: 2
For any b > a > 0, consider the probability density function (pdf)...
 2x
0 xa
 ab ,

 2( x  b)
f ( x)  
, a xb
 b( a  b)

0,
else
Determine the cumulative distrib function (cdf)
F ( x)  P( X  x)
For any x  0, it follows that F ( x)  0.
For any 0  x  a, it follows that
x
F ( x)  0   2t dt 
0 ab
2( x  b)
b(a  b)
2x
ab
For any a  x  b, it follows that
F ( x)  F (a )  
x
a
2(t b)
dt 
b(ab)
1
(b)  0
For any x  b, it follows that F ( x)  F
a
0
x
b
x
32
Four Examples: 2
For any b > a > 0, consider the probability density function (pdf)...

 2x
  Edistrib
[ X ] function
x f ((cdf)
x) dx
Determine the mean
cumulative

,
0

x

a

 ab

a
b
 2( x  b)
 x f ( x) dx  x f ( x ) dx
f ( x)  
, a xb
0
a
b
(
a

b
)

a 2x
b 2( x  b)

 x dx  x
dx 
0
a
0,
ab
b(a  b)
else




Determine the variance
2( x  b)
b(a  b)
2x
ab
2

0
a

  E  X   E  X    x 2 f ( x) dx   2

2
a
0
2
b
2x
2 2( x  b)
x
dx   x
dx   2 
a
ab
b(a  b)
2
b
x
33
Four Examples: 3
Consider the following probability density function (pdf)...
2
 , x 1
f ( x)   x3
0,
x 1
Confirm pdf



f ( x) dx  

1

  E[ X ]  


x f ( x) dx

 2
2
   x 3 dx   2 dx
1
1 x
x
1 c
c
x 
 lim 2 x 2 dx  lim 2  
1
c 
c 
 1  1
c
c
c
x 
c
2
3
dx  lim 2  x dx  lim 2 

3
c 
1
c 

2
x
 1
c
1
1


 lim   2   1  lim 2  1
c 
c  c
 x 1
2
2
 2
 lim     2  lim  2
c 
c  c
 x 1
35
Four Examples: 4
3
Consider the following probability density function (pdf)...
 12
 , x 1
f ( x)   x 23
0,
x 1
Confirm pdf



f ( x) dx  

1
1 2 c c
 x x  
c c
12
23
dx  lim 2  x x dxdx  lim 2   
32
c 
c  1 1
x

 12 1 1
cc
1
1
11
 
 lim   2  11lim
lim 2 1 1
c 
c 
c cc
x
  1 1

  E[ X ]  


x f ( x) dx

 1
21
2
   x 32 dx   2 dx
dx
1
1
x
x
1 c
c c
x 
2
 lim 2  x x1dx
dx  lim 2  
c  1 1
c 
 1  1
c
c
2
 2
 lim     2  lim  2
c 
c  c
 x 1
36
Four Examples: 4
3
Consider the following probability density function (pdf)...
 12
 , x 1
f ( x)   x 23
0,
x 1
Confirm pdf



f ( x) dx  

1
1 2 c c
 x x  
c c
12
23
dx  lim 2  x x dxdx  lim 2   
32
c 
c  1 1
x

 12 1 1
cc
1
1
11
 
 lim   2  11lim
lim 2 1 1
c 
c 
c cc
x
  1 1

  E[ X ]  


1


x f ( x) dx
 1
1
x 2 dx  
dx
1
x
x
 lim  x 1 dx  lim  ln | x |1
c
c 
1
 lim (ln c)
c 
c
c
c 
 
37