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Discussion 11A
Continuous Distributions
PDF
We describe the distribution of a continuous random variable with a
probability density function
f
(
x
)
f(x)
f
(
x
)
such that
Pr
[
X
∈
[
a
,
b
]
]
=
∫
a
b
f
(
x
)
d
x
\Pr[X \in [a, b]] = \int_a^b f(x) dx
Pr
[
X
∈
[
a
,
b
]
]
=
∫
a
b
f
(
x
)
d
x
.
f
(
x
)
f(x)
f
(
x
)
satisfies
∫
−
∞
∞
f
(
x
)
d
x
=
1
\int_{-\infty}^\infty f(x) dx = 1
∫
−
∞
∞
f
(
x
)
d
x
=
1
.
f
(
x
)
f(x)
f
(
x
)
can be thought of as roughly
Pr
[
X
∈
[
x
,
x
+
d
x
]
]
=
f
(
x
)
d
x
\Pr[X \in [x, x+dx]] = f(x) dx
Pr
[
X
∈
[
x
,
x
+
d
x
]
]
=
f
(
x
)
d
x
.
The word
“density”
fits our physical interpretation of distributions from before.
CDF
It can be useful to describe a continuous random variable with a
cumulative distribution function:
F
(
x
)
=
∫
−
∞
x
f
(
x
)
d
x
=
Pr
[
X
≤
x
]
F(x) = \int_{-\infty}^x f(x) dx = \Pr[X \leq x]
F
(
x
)
=
∫
−
∞
x
f
(
x
)
d
x
=
Pr
[
X
≤
x
]
.
Using the CDF it is easy to compute
Pr
[
X
∈
[
a
,
b
]
]
=
F
(
b
)
−
F
(
a
)
\Pr[X \in [a, b]] = F(b) - F(a)
Pr
[
X
∈
[
a
,
b
]
]
=
F
(
b
)
−
F
(
a
)
.
Analogies
Definitions for discrete random variables have analogous definitions for continuous random variables: Bayes’, Total Probability, etc.
E
[
X
]
=
∫
−
∞
∞
x
f
(
x
)
d
x
E[X] = \int_{-\infty}^\infty x f(x) dx
E
[
X
]
=
∫
−
∞
∞
x
f
(
x
)
d
x
E
[
g
(
X
)
]
=
∫
−
∞
∞
g
(
x
)
f
(
x
)
d
x
E[g(X)] = \int_{-\infty}^\infty g(x) f(x) dx
E
[
g
(
X
)
]
=
∫
−
∞
∞
g
(
x
)
f
(
x
)
d
x
Pr
[
q
(
X
,
Y
)
]
=
∫
Pr
[
q
(
X
,
y
)
∣
Y
=
y
]
f
Y
(
y
)
d
y
\Pr[q(X,Y)] = \int \Pr[q(X,y) \mid Y=y] f_Y(y) dy
Pr
[
q
(
X
,
Y
)
]
=
∫
Pr
[
q
(
X
,
y
)
∣
Y
=
y
]
f
Y
(
y
)
d
y
Exponential Distribution
X
X
X
denotes the amount of time in order for an event to occur, where the event occurs at a rate of
λ
\lambda
λ
t
i
m
e
−
1
\text{time}^{-1}
time
−
1
.
f
X
(
x
)
=
λ
e
−
λ
x
f_X (x) = \lambda e^{-\lambda x}
f
X
(
x
)
=
λ
e
−
λ
x
for
x
≥
0
x \geq 0
x
≥
0
.
E
[
X
]
=
1
λ
E[X] = \frac{1}{\lambda}
E
[
X
]
=
λ
1
V
a
r
(
X
)
=
1
λ
2
\mathrm{Var}(X) =\frac{1}{\lambda^2}
V
a
r
(
X
)
=
λ
2
1
F
(
X
)
=
1
−
e
−
λ
x
F(X) = 1 - e^{-\lambda x}
F
(
X
)
=
1
−
e
−
λ
x
The exponential distribution is memoryless, so
Pr
[
X
>
t
]
=
Pr
[
X
>
t
+
s
∣
X
>
s
]
\Pr[X > t] = \Pr[X > t + s \mid X > s]
Pr
[
X
>
t
]
=
Pr
[
X
>
t
+
s
∣
X
>
s
]
.
It can be thought of as geometric distribution, where
λ
\lambda
λ
triggers occur every small time interval.
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Continuous Distributions
PDF
CDF
Analogies
Exponential Distribution