Statistics
Lecture notes
Lecture 1
Probability theory = the consequences of the presence of chance are considered -> so the first thing
needed is chance
- Random experiment = an experiment for which the actual outcome is determined by chance
- Possible outcomes = most basic possible results of a random experiment
- Sample space (Ω) = set of all N possible outcomes
o Events (A, B, …) are subsets of the sample space
Single events / multiple events / empty event (Ø)
o Complement = part of the event of A
A venn diagram = a useful method to represent the sample space. For example experiment is
tossing a coin, possible outcomes are (Head (H), Tail (T), TT, HT, TH). The venn diagram will
look like this;
A ᴗ B = the union of A and B -> A or B occurs
A ᴖ B = the intersection of A and B -> A and B both occur
A ᴖ B = Ø -> means that A and B are disjoint -> they do not have common outcomes
Probability measure is also called the model P, this assigns real numbers P(A) to all events of A, in
such a way that it is possible to say;
- P(A) ≥ 0
- P(Ω) = 1
- P(A ᴗ B) = P(A) + P(B) -> if A, B disjoint
A requirement in probability is that all outcomes are equally likely
P(S) = N(A) / N -> where N(A) = # of outcomes in A
This type of random experiment is like throwing a fair die with N sides
Conditional probability -> focuses on the probability of event A given that event B occurs. Is denoted
as;
P(A | B) = P(A ᴖ B) / P(B)
Here P(B) >0
A and B are independent, if the fact that throwing B (occurrence of B) does not influence the
probability of A. so this is denoted as = P(A |B) = P(A)
, Random variables = a prescript that attaches a value to each outcome of the sample space. It is a
quantity for which the actual outcome is determined by chance -> the outcome of a RV is called the
realisation. You can have 2 types of RV;
- Quantitative = values are ordinary numbers
- Qualitative = values are categories
Quantitative random variables are called the outcome. These RV’s can be either discrete or
continuous
1) Discrete = finite or countable number of outcomes. examples = number of children in
household, number of rounds needed before a game is over, number of correct answers in
MC test (all round numbers, no decimal)
2) Continuous = all values in an interval are possible outcomes. examples = next week sales in
store, next quarter inflation rate, time to solve a puzzle, waiting time for the train
For a discrete random variable, you apply the probability density function (pdf) f of a discrete RV X,
this is defined by;
f(x) = P(X = x) for all outcomes x of X
f(x) is the probability that the realisation of X will be x
cumulative distribution function (cdf) F of an RV X is defined by;
F(a) = P(X ≤ a) for all real numbers of a
Properties that belong to the cdf are;
- F is non-decreasing
- F (-∞) = 0 and F(∞) = 1
- F (b) – F(a) = P(a < X ≤ b) for all a and b, with a > b
The expected value or mean determines the expected value of X. Can be noted as μ or μor or μμx. or μThe
expected value (E(X)) is a weighted average of the outcomes of X, f(x) is the weight of the outcome x
If you have a function following from f(x) -> for instance f(v), then you need to calculate both to find
the expected value. Which is done almost the same as E(X)
Variation or o2 is the expectation of the squared random deviation. Variance is always non-negative
-> V(X) ≥ 0
- V(a) = 0, for each constant a
- Idea behind the variation is that you take the random deviation X – μ or μand or μthen or μsquare or μit
Standard deviation of X, you take the square root of the variance
Lecture notes
Lecture 1
Probability theory = the consequences of the presence of chance are considered -> so the first thing
needed is chance
- Random experiment = an experiment for which the actual outcome is determined by chance
- Possible outcomes = most basic possible results of a random experiment
- Sample space (Ω) = set of all N possible outcomes
o Events (A, B, …) are subsets of the sample space
Single events / multiple events / empty event (Ø)
o Complement = part of the event of A
A venn diagram = a useful method to represent the sample space. For example experiment is
tossing a coin, possible outcomes are (Head (H), Tail (T), TT, HT, TH). The venn diagram will
look like this;
A ᴗ B = the union of A and B -> A or B occurs
A ᴖ B = the intersection of A and B -> A and B both occur
A ᴖ B = Ø -> means that A and B are disjoint -> they do not have common outcomes
Probability measure is also called the model P, this assigns real numbers P(A) to all events of A, in
such a way that it is possible to say;
- P(A) ≥ 0
- P(Ω) = 1
- P(A ᴗ B) = P(A) + P(B) -> if A, B disjoint
A requirement in probability is that all outcomes are equally likely
P(S) = N(A) / N -> where N(A) = # of outcomes in A
This type of random experiment is like throwing a fair die with N sides
Conditional probability -> focuses on the probability of event A given that event B occurs. Is denoted
as;
P(A | B) = P(A ᴖ B) / P(B)
Here P(B) >0
A and B are independent, if the fact that throwing B (occurrence of B) does not influence the
probability of A. so this is denoted as = P(A |B) = P(A)
, Random variables = a prescript that attaches a value to each outcome of the sample space. It is a
quantity for which the actual outcome is determined by chance -> the outcome of a RV is called the
realisation. You can have 2 types of RV;
- Quantitative = values are ordinary numbers
- Qualitative = values are categories
Quantitative random variables are called the outcome. These RV’s can be either discrete or
continuous
1) Discrete = finite or countable number of outcomes. examples = number of children in
household, number of rounds needed before a game is over, number of correct answers in
MC test (all round numbers, no decimal)
2) Continuous = all values in an interval are possible outcomes. examples = next week sales in
store, next quarter inflation rate, time to solve a puzzle, waiting time for the train
For a discrete random variable, you apply the probability density function (pdf) f of a discrete RV X,
this is defined by;
f(x) = P(X = x) for all outcomes x of X
f(x) is the probability that the realisation of X will be x
cumulative distribution function (cdf) F of an RV X is defined by;
F(a) = P(X ≤ a) for all real numbers of a
Properties that belong to the cdf are;
- F is non-decreasing
- F (-∞) = 0 and F(∞) = 1
- F (b) – F(a) = P(a < X ≤ b) for all a and b, with a > b
The expected value or mean determines the expected value of X. Can be noted as μ or μor or μμx. or μThe
expected value (E(X)) is a weighted average of the outcomes of X, f(x) is the weight of the outcome x
If you have a function following from f(x) -> for instance f(v), then you need to calculate both to find
the expected value. Which is done almost the same as E(X)
Variation or o2 is the expectation of the squared random deviation. Variance is always non-negative
-> V(X) ≥ 0
- V(a) = 0, for each constant a
- Idea behind the variation is that you take the random deviation X – μ or μand or μthen or μsquare or μit
Standard deviation of X, you take the square root of the variance
