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Discussion 9B
Random Variables
A random variable is an assignment of a number to each sample point of an experiment.
The distribution of a
(discrete)
random variable
X
X
X
is given by the values
Pr
[
X
=
x
]
\Pr[X=x]
Pr
[
X
=
x
]
for all values of
x
x
x
, where
Pr
[
X
=
x
]
=
∑
ω
:
X
(
ω
)
=
x
Pr
[
ω
]
\Pr[X=x] = \sum_{\omega : X(\omega) = x} \Pr[\omega]
Pr
[
X
=
x
]
=
∑
ω
:
X
(
ω
)
=
x
Pr
[
ω
]
.
Bernoulli/Indicator Random Variables
X
∼
B
e
r
n
o
u
l
l
i
(
p
)
X \sim \text{Bernoulli}(p)
X
∼
Bernoulli
(
p
)
Pr
[
X
=
1
]
=
p
\Pr[X=1] = p
Pr
[
X
=
1
]
=
p
and
Pr
[
X
=
0
]
=
1
−
p
\Pr[X=0] = 1-p
Pr
[
X
=
0
]
=
1
−
p
Denotes outcome of a binary event
(coin
flips) where
p
p
p
is the probability of a success
Geometric Random Variables
X
∼
G
e
o
m
e
t
r
i
c
(
p
)
X \sim \text{Geometric}(p)
X
∼
Geometric
(
p
)
Pr
[
X
=
x
]
=
(
1
−
p
)
x
−
1
p
\Pr[X=x] = (1-p)^{x-1} p
Pr
[
X
=
x
]
=
(
1
−
p
)
x
−
1
p
Denotes number of independent trials until a success
(number
of flips until heads) where
p
p
p
is the probability of a success
Binomial Random Variables
X
∼
B
i
n
o
m
i
a
l
(
n
,
p
)
X \sim \text{Binomial}(n, p)
X
∼
Binomial
(
n
,
p
)
Pr
[
X
=
x
]
=
(
n
x
)
p
x
(
1
−
p
)
n
−
x
\Pr[X=x] = \binom{n}{x} p^x (1-p)^{n-x}
Pr
[
X
=
x
]
=
(
x
n
)
p
x
(
1
−
p
)
n
−
x
Denotes the number of successes in
n
n
n
independent trials where
p
p
p
is the probability of a success
Expectation
E
[
X
]
=
∑
x
x
Pr
[
X
=
x
]
E[X] = \sum_{x} x \Pr[X=x]
E
[
X
]
=
∑
x
x
Pr
[
X
=
x
]
Linearity of Expectation
E
[
X
+
a
Y
]
=
E
[
X
]
+
a
E
[
Y
]
E[X + a Y] = E[X] + a E[Y]
E
[
X
+
a
Y
]
=
E
[
X
]
+
a
E
[
Y
]
X
∼
B
e
r
n
o
u
l
l
i
(
p
)
X \sim \text{Bernoulli}(p)
X
∼
Bernoulli
(
p
)
E
[
X
]
=
p
E[X] = p
E
[
X
]
=
p
X
∼
G
e
o
m
e
t
r
i
c
(
p
)
X \sim \text{Geometric}(p)
X
∼
Geometric
(
p
)
E
[
X
]
=
1
p
E[X] =\frac{1}{p}
E
[
X
]
=
p
1
X
∼
B
i
n
o
m
i
a
l
(
n
,
p
)
X \sim \text{Binomial}(n, p)
X
∼
Binomial
(
n
,
p
)
E
[
X
]
=
n
p
E[X] = np
E
[
X
]
=
n
p
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Random Variables
Bernoulli/Indicator Random Variables
Geometric Random Variables
Binomial Random Variables
Expectation
Linearity of Expectation