Section 6.1: Discrete Random Variables
A random variable
is discrete if and only if its set of possible
values is finite or, at most, countably infinite
➤
A discrete random variable
is uniquely determined by
Its set of possible values
Its probability density function (pdf):
A real-valued function
defined for each
probability that has the value
➤
✦
as the
✦
By definition,
Section 6.1: Discrete Random Variables
1
is Equilikely
and each possible value is equally likely
is the sum of the two up faces,
If
Example 6.1.2 Roll two fair dice
➤
Example 6.1.1
➤
Examples
From example 2.3.1,
Section 6.1: Discrete Random Variables
2
Example 6.1.3
A coin has
as its probability of a head
➤
Toss it until the first tail occurs
➤
If
:
Verify that
and the set of possible values is infinite
&
%
$
"
#
!
➤
is Geometric
➤
and the pdf is
is the number of heads,
➤
Section 6.1: Discrete Random Variables
3
Cumulative Distribution Function
is Equilikely
If
then the cdf is
.
.
is Geometric
If
then the cdf is
"
)
#
01
/ '
)
➤
"
-
)
'
➤
)
*
+, (
'
The cumulative distribution function(cdf) of the discrete random
for each
as
variable is the real-valued function
' ➤
Section 6.1: Discrete Random Variables
4
62
67
5
8
4
2
62
3
7
9 7
?
8 <=
9 7
>
Section 6.1: Discrete Random Variables
67
5
8
4
2
3
7
:97
8=
9:7
6
;<
2
is small enough to tabulate the cdf
:97
➤
for sum of two dice
9 6
7
No simple equation for
➤
'
Example 6.1.5
5
Relationship Between cdfs and pdfs
=
'
➤
'
'
A pdf can be generated from its corresponding cdf by subtraction
➤
'
'
' For example,
A cdf can be generated from its corresponding pdf by recursion
➤
A discrete random variable can be defined by specifying either its
pdf or its cdf
Section 6.1: Discrete Random Variables
6
Other cdf Properties
1 @
'
$
The cdf values are bounded between 0.0 and 1.0
Monotonicity of
is the basis to generate discrete random
variates in the next section
➤
'
➤
, then
$
1
if
'
A cdf is strictly monotone increasing:
@
➤
Section 6.1: Discrete Random Variables
7
Mean and Standard Deviation
➤
The corresponding standard deviation
is
is
➤
The variance is
A$
$
B
A
$
or
B $
B
D E
F
D C
B
A
of the discrete random variable
The mean
A
➤
Section 6.1: Discrete Random Variables
8
$
1$ %H
B
G
and
Section 6.1: Discrete Random Variables
.
G
I
M
G
$
B
"
$
A
and
A
$
D K
L
D C
1$
is the sum of two dice then
"
If
,
1$
➤
is Equilikely
When
A
A
and
B
then the mean and standard deviation are
I
J
is Equilikely
If
➤
Examples
9
is Geometric
If
then the mean and standard deviation are
1
1
"
"
#
$
$
A $
$
B $
$
!
"
!
"
#
A
!
!
➤
Another Example
B
N
$
B $
..
Section 6.1: Discrete Random Variables
10
Expected Value
The expected value of the discrete random variable
A
is
➤
Q
The mean of a random variable is also known as the expected value
OP
➤
W
V
A
U
T
U
SR
Section 6.1: Discrete Random Variables
) is the mode, which can
The most likely value (with largest
be different from the expected value
➤
$
1
Q
Expected value refers to the expected average of a large sample
corresponding to :
as
.
OP
➤
11
➤
The most likely number of heads is 0
➤
The expected number of heads is 1
➤
0 occurs with probability
and 1 occurs with probability
M
Toss a fair coin until the first tail appears
.
➤
.
Example 6.1.10
The most likely value is twice as likely as the expected value
➤
For some random variables, the mean and mode may be the same
For the sum of two dice, the most likely value and expected value
are both 7
Section 6.1: Discrete Random Variables
12
[
X
Section 6.1: Discrete Random Variables
Q
X
Note: in general, this is not equal to
OP
Q
OP
X
is
\Q
The expected value of
[
U
is a new random variable, with possible values
OP
➤
for all possible values of
\
➤
\
X
X Define function
X ZY
➤
More on Expectation
13
➤
$
B $
A
$
B $
A $
$
Q
OP
with equality if and only if
Section 6.1: Discrete Random Variables
$
A $
$
Q$
OP
B $
Q
OP
^_Q $
OP
$
OP
\Q
OP
A $
Q
A $
$
,
]
Q$
Q
\Q
OP
A
]
,
A
So that
➤
If
A
➤
with
OP
If
$
➤
OP
Example 6.1.11
is not really random
14
Example 6.1.12
➤
Q
OP
Q
and ,
for constants
OP
\
\Q
If
OP
➤
Suppose
is the number of heads before the first tail
Win $2 for every head and let be the amount you win
\
✦
✦
The possible values
➤
Your expected winnings are
Section 6.1: Discrete Random Variables
Q
OP
Q
OP
\Q
OP
]
you win are defined by
X
\
➤
15
Discrete Random Variable Models
Section 6.1: Discrete Random Variables
For example, the functions Equilikely and Geometric generate
and Geometric
random variates corresponding to Equilikely
random variables, respectively
➤
A random variate is an algorithmically generated possible value of
a random variable
➤
A random variable is an abstract, but well defined, mathematical
object
➤
16
➤
The cdf:
➤
The mean:
➤
The variance:
➤
The standard deviation:
$
B
$
`1
B $
A
for
/`1
'
Section 6.1: Discrete Random Variables
for
The pdf:
➤
with probability
and
with probability
➤
with possible values
The discrete random variable
➤
Bernoulli Random Variable
17
To generate a Bernoulli
➤
Bernoulli Random Variate
random variate
if (Random()< 1.0-p)
return 0;
else
return 1;
V
Monte Carlo simulation that uses
replications to estimate an
unknown probability is equivalent to generating an iid sequence
random variates
of Bernoulli
V
➤
Section 6.1: Discrete Random Variables
18
Example 6.1.14
➤
➤
Pick-3 Lottery: pick a 3-digit number between 000 and 999
Costs $1 to play the game and wins $500 if a player matches the
3-digit number chosen by the state
➤
The player’s expected yield is
be the player’s yield
#
G
X
X
X
\Q
OP
1
b
bM
X
a
\
Let
X
➤
Section 6.1: Discrete Random Variables
19
as its probability of a head and toss this coin
V
is a Binomial
times
random
and the pdf is
V
R
`
V
V
➤
Let
be the number of heads;
variable
➤
A coin has
➤
V
Binomial Random Variable
tosses of the coin generate an iid sequence
random variables and
Bernoulli
Section 6.1: Discrete Random Variables
R
1
$
R of
$
1
V
➤
20
Binomial equation
➤
In the particular case where
j
i
c
R
`
R
"
#
V
R
➤
d
fge
h
Verify that
R
`
"
#
R
R
V
R
and
Section 6.1: Discrete Random Variables
21
"
#
Section 6.1: Discrete Random Variables
V
A $
+
o
V
V
q
V
q
`q)
)
po
q
V
p
+
and
$
The variance is
R
p
# +
o
"
1
V
o
o
`
V
1
R
`
V
o
R
"
"
#
#
Q
OP
R
`
A
V
R
R
The mean is
A $
➤
V
Let
OP
"
)
➤
Q$
B $
A
n h
mlgke
Mean and Variance of Binomial
22
Pascal Random Variable
A coin has as its probability of a head and toss this coin until the
tail occurs
V )
r
➤
is the number of heads,
is a Pascal
V
If
random variable
={0,1,2,...} and the pdf is
➤
R
V
➤
Section 6.1: Discrete Random Variables
23
Pascal Random Variable ctd.
V
$
V
`
R
V
Negative binomial expansion:
➤
Prove that the infinite pdf sum converges to 1
➤
It can also be shown that
`
R
R
"
#
$
A $
$
A $
Q$
OP
B $
V
!
"
#
Q
A
OP
V
!
"
#
R
V
!
➤
Section 6.1: Discrete Random Variables
24
X
€" x vu XX
 XX
+
w X
%
V
We see that a Pascal
random variable is the sum of iid
random variables
Geometric
➤
$
1
&
|z" x vu XX
} XX
X
+X
w
t
~z" x uv XX
X
& w +X
t X
X
is large, a head/tail sequence might be
X
yz" x vu X
X
{ X
+
w X
t X
V
and
X
For example,if
t X
➤
M
V
R
$
1
V
s
If
and
is an iid sequence of Geometric
random variables, the sum is a Pascal
random variable
V
➤
Example 6.1.17
Section 6.1: Discrete Random Variables
25
‰
Section 6.1: Discrete Random Variables
!
R
V
ˆ
‰
A
† …„
and
A
A
So that
A
A
‚
ƒR
V
V
V
A
o
o
V
V
o
`
ƒR
‚
ƒ
‚
A
ƒR
‚
Vo
A
A
A
Vo
W
V
U
as
V
Vo
and
V
o
A
Fix
A
➤
o
Poisson
o
ˆ
‡†…„
V
ˆ
‡†…„
➤
!
R
!
R
is a limiting case of Binomial
V
A
.
V
Poisson Random Variable
It can be shown that
26