probability Flashcards
U
union
∩
intersection
∅
empty set
−
difference
⊆, ⊂
subset
∈, ∉
contains
| |
|
number of elements
[ ], ( )
set limits
∀
for all
probability
A number between 0 and 1, describing the relative possibility (chance of likelihood) that an event will occur
small probability
unlikely
high probability
likely
A random variable X
a variable which can take a value out of a set of values, due to chance, unobserved factors or measurement errors
Probability P
the chance or likelihood that we observe something
sample space S
the set of all possible outcomes
event E ⊆ S
a collection of one or more outcomes
possible outcome x ∈ S
one of the values that X can take
union of events: A U B
all possible outcomes in A or in B
intersection of events: A n B
all possible outcomes in A and in B
difference of events: A – B
all possible outcomes in A that are not in B
complementary event A’ = S – A
All possible outcomes that are not in A
probability of an event
number of favourable outcomes divided by the number of possible outcomes
categorical variable
random variable that represents a quality or category
numerical variable
a random variable that can be measured.
dichotomous or binary variable
it can only take two variables (yes or no)
nominal variable
there is no order and it is mutually exclusive (A-B-C)
ordinal variable
it can be ordered and it is mutually exclusive and “Quasi-quantitative” (I-II-III)
discrete variable
the sample space is countable, there is a finite number of values (2-4-6)
continuous variable
the sample space is uncountable, between a min and max value (2,3-3,5-4,6)
conditional probability
the probability of A, given that B happened
Bayes theorem
P(A|B) = P(A n B)/P(B)
two interdependent events A, B ⊆ S
P(A n B) = P(A)P(B)
uniform distribution (discrete)
all values have the same probability (eg dice)
bernouilli distribution (discrete)
binary variables. Success and Failure
Binomial distribution (discrete)
the number of successes in n Bernouilli runs with probability p
poisson distribution (discrete)
- k ∈ {0,1,2,…} is the number of times an event occurs in an interval
- events are independent
- the average rate at which events occur is independent if any occurences
- two events cannot occure at exactly the same instant
uniform distribution (continuous)
The outcomes are equally likely
normal (gaussian) distribution (continuous)
it is symmetric at the mean, showing that the data close to the mean is more frequent than far from from the mean
central limit theorem
in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution.
beta distribution (continuous)
this is often used to model the uncertainty about the probability of success of an experiment
chi-square ditribution (continuous)
it is a distribution with k degrees of freedom. It is used to describe the distribution of a sum of squared random variables.
lognormal distribution
eg the income of 97%-99% of the population, city sizes
pareto distribution (80-20 rule)
eg income, sizes of human settlements, values of oil reserves in oil fields, standardized price returns on individual stocks
negative binomial distribution
number of failures before a given number of successes in a binomial
exponential distribution
time between two consecutive next Poisson-type events
gamma distribution
time before the next k Poisson-type events occur
