1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
|
\title{Noter til probability m2}
\section{Random Variables}
Her mapper man fra et sample space S til en variabel.
Her kalder man variablen et stort tal R eller sådan noget.
Derfor er et random variabel egentlig en transformation mellem S og real tal.
\emph{X er en descrete random variable hvis dens range er countable.}
For continues random variables the following is true:
$$
P(X = x) = 0
$$
\subsection{Cumulative Distribution Function}
Her måler man prob for at ens random er mindre end et bestemt tal.
$$
F(x) = P(X \leq x)
$$
Man kan også finde det for en range:
$$
P(a < X \leq b) = F(b) - F(a)
$$
Ved discrete random variables vil denne være en slags trappe.
Kan sige at den er \emph{continues from the right} eftersom man har $\leq$ i definition.
\subsection{Probability Mass Function}
Works only for discrete random variables.
Is defines as the probability that $X = a$:
$$
p(a) = P(X = a)
$$
From here CDF can be found:
$$
F(a) = \sum_{all x \leq a} p(a)
$$
\subsection{Probability Density Function}
Her finder man P i et evigt lille interval:
Is the derivative of the CDF.
$$
F(a) = P(X \in (-\infty,a]) = \int_{-\infty}^a f(x) dx \\
f(a) = \frac{d}{da} F(a)
$$
The following must be true:
$$
\int_{-\infty}^{\infty} f(x) dx = 1
$$
\subsection{Multiple Random Variables}
Have multiple random variables, which can be or is not correlated.
Can define the joined CDF:
$$
F_{XY}(x,y) = P(X \leq x, Y \leq y)
$$
One can also find the probability of one of the variables. (The \emph{marginal})
$$
F_X(x) = P(X \leq x) = P(X \leq, Y < \infty) = F(x, \infty)
$$
One can not go from marginal to the joined, as they do not contain enough information.
This is only possible if X and Y are \emph{independent}.
$$
F_{XY}(x,y) = F_X(x) \cdot F_Y(x) \\
p(x,y) = p_X(x) \cdot p_Y(y) \\
f(x,y) = f_X(x) \cdot f_Y(y)
$$
\subsection{Joined PMF}
$$
P_{XY}(x,y) = P(X = x, Y = y)
$$
% vim: spell spelllang=da,en
|
