infinitely independent tosses of a coin with
P(Heads) =
p
Sample space:
infinite sequences of H and T
Random variable
X
:
number of tosses until the first
Heads
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Bernoulli
Uniform
Binomial
Geometric
Expectation
12/23
Geometric Random Variable
Geometric with parameter
0
< p
≤
1
Experiment:
infinitely independent tosses of a coin with
P(Heads) =
p
Sample space:
infinite sequences of H and T
Random variable
X
:
number of tosses until the first
Heads
Geometric random variable
p
X
(
k
) = (1

p
)
k

1
p
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
13/23
5min Break!
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
14/23
1
Discrete Random Variables
2
Common Random Variables
3
Expectation
The Mean
Expected Value Rule
The Variance
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
15/23
The Mean
Mean
The
expected value
( also called the expectation or the mean)
of a random variable
X
, with PMF
p
X
, is defined by
E
[
X
]
,
X
x
xp
X
(
x
)
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
16/23
Bernoulli Random Variable
Bernoulli random variable
Bernoulli with parameter
p
∈
[0
,
1]
X
=
(
1
w.p.
p
0
w.p.
1

p
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
17/23
Uniform Random Variable
Consider a uniform random variable on
[0
,
1
, . . . , n
]
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
18/23
Linearity of Expectations
Linearity of Expectations
Given a random variable
X
:
E
[
aX
+
b
] =
a
E
[
X
] +
b
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
19/23
Expected Value Rule
Let
X
be a random variable, and
Y
=
g
(
X
)
be a function of
X
. By definition:
E
[
g
(
X
)] =
E
[
Y
] =
X
y
yp
Y
(
y
)
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
19/23
Expected Value Rule
Let
X
be a random variable, and
Y
=
g
(
X
)
be a function of
X
. By definition:
E
[
g
(
X
)] =
E
[
Y
] =
X
y
yp
Y
(
y
)
Expected Value Rule
E
[
g
(
X
)] =
X
x
g
(
x
)
p
X
(
x
)
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
20/23
The Variance
Average distance from the mean?
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
20/23
The Variance
Average distance from the mean?
Variance
Variance is the expected value of the random variable
g
(
X
) = (
X

E
[
X
])
2
:
var
(
X
) =
E
[(
X

E
[
X
])
2
]
Probability &
Statistics
Discrete
Random
Variables
Common
Random
Variables
Expectation
The Mean
Expected Value
Rule
The Variance
20/23
The Variance
Average distance from the mean?
You've reached the end of your free preview.
Want to read all 52 pages?
 Fall '14
 LuisDavidGarciaPuente
 Probability theory, Nguyen Minh Huong