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4. Statistics and Probability
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4. Statistics and Probability
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4. Statistics and Probability
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Modeling is a process of creating a mathematical representation of certain real-world scenarios. This falls under the
statistics
field
,
which requires some terminology in prior.
4.1 Statistics
There are two types of probability,
experimental
and
theoretical.
4.2 Probability
Discrete Distributions
If
X
X
X
is a random variable with sample space {
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
} and corresponding probabilities
p
i
p_i
p
i
, then {
p
1
,
p
2
,
.
.
.
,
p
n
p_1, p_2, ..., p_n
p
1
,
p
2
,
...
,
p
n
} describes the probability distribution of
X
X
X
. We call
P
(
x
)
=
P
(
X
=
x
)
P(x) = P(X=x)
P
(
x
)
=
P
(
X
=
x
)
as the probability mass function (PMF).
Continuous distributions
A random variable
X
X
X
is continuous if its cumulative distribution function is continuous. If we have: (HL)
F
X
(
x
)
=
∑
−
∞
x
f
X
(
t
)
d
t
F_X(x)=\sum _{-∞}^xf_X(t) dt
F
X
(
x
)
=
∑
−
∞
x
f
X
(
t
)
d
t
f
X
f_X
f
X
is the probability density function (PDF) of
X
X
X
. Note that PDF and PMF are fundamentally different objects, as PDFs can be greater than 1. Below, for simplicity, we denote PDF as
f
(
x
)
f(x)
f
(
x
)
.
One example of such a distribution is the normal distribution. (SL)
4.3 Distributions
4.4 Random variables
done
4.4 Random Variables
4.4 Random Variables
