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Summary Introduction Statistics
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Here is a summary of the theory covered during the course Introduction to Statistics.
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Chapter 1 What Is Statistics?The mean of a sample n measured responses y1, y2, …, yn is given by
n
1
y=
n ∑ yi
i=1
The corresponding population mean is denoted μ .
The variance of a sample of measurements y1, y2, …, yn is the sum of the square
of the differences between the measurements and their mean divided by n – 1
.Symbolically, the sample variance is
y i− y ¿2
¿
n
1
n−1 ∑
2
s= ¿
i=1
The corresponding population variance is denoted by the symbol σ 2
.
The standard deviation of a sample of measurements is the positive square root
of the variance; that is, s= √ s2 .
The corresponding population standard deviation is denoted by σ =√ σ 2 .
Empirical Rule
For a distribution of measurements that is approximately normal (bell shaped), it
follows that the interval with end points
μ ±σ contains approximately 68% of the measurements.
μ ±2 σ contains approximately 95% of the measurements.
μ ±3 σ contains almost all of the measurements.
Chapter 2 Probability
The union of A and B, denoted by U ∪ A, is the set of all point in A or B or
both.
The intersection of A and B, denoted by A ∩ B or by AB, is the set of all point
in both A and B.
The complement of A (AC) is the set of points that are in S but not in A.
Two sets, A and B, are said to be disjoint, or mutually exclusive, if A ∩ B=
∅
Distributive laws:
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C),
, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C),
DeMorgan’s laws:
(A ∩ B)C = AC ∪ BC and (A ∪ B) C = AC ∩ BC
A simple event is an event that cannot be composed. Each simple event
corresponds to one and only one sample point. The letter E with a subscript will
be used to denote a simple event or the corresponding sample point.
The sample space associated with an experiment is the set consisting of all
possible sample points. The set of all possible outcomes of an experiment.
Denoted by S.
A discrete sample space is one that contains either a finite or a countable
number of distinct sample points.
To every event A in S, we assign an number, P(A), called the probability of A, so
that the following axioms hold:
Ax 1: P(A) ≥ 0 for all A ⊂ B
Ax 2: P(S) = 1
Ax 3: If A1, A2, A3, … form a sequence of pairwise mutually exclusive events in S
(that is Ai ∩ Aj = ∅ if
i ≠ j), then
∞
P(A1 ∪ A2 ∪ A3 ∪ …) = ∑ P( A i)
i=1
If a sample space contains N equiprobable sample points and an event A contains
exactly na sample points, it is easily seen that P(A) = n a / N
With m elements a1, a2, …, am and n elements b1, b2, …., bn, it is possible to form
mn = m x n pairs containing one element from each group.
Permutation (met volgorde en zonder terugleggen)
An ordered arrangement of r distinct objects is called a permutation. The number
of ways of ordering n distinct object taken r at a time will be designated by the
symbol Pnr .
n!
Pnr =n ( n−1 ) ( n−2 ) … ( n−r +1 )=
(n−r )!
Multinomial (n objecten verdelen over k verschillende groepen (volgorde
binnen groep maakt niet uit))
The number of ways partitioning n distinct object into k distinct groups containing
n1, n2, …, nk object, respectively, where each object appears in exactly one group
k
and ∑ ¿=n .
i=1
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