Probability
Long-run Frequency Interpretation
example: "The probability of an event is the frequency of it occurring
in
the long run"
Subjective Interpretation
example: "The probability of an event is an expression of
how
likely one thinks its occurrence is"
Mathematical:
example: "Probability is any function with arguments in the sample
space
and range in the interval (0,1) which follows the probability calculus"
Random Variables and Probability:
A random experiment is a set
up with definite and identifiable
outcomes, none of which can be predicted exactly.
Events are collections of the
possible outcomes of a random
experiment. The sample
space is the collection of
all possible events (which we will call S). A
probability function P is a function
that maps elements of the sample space into the interval (0,1) with the
following properties:
(i) The probability of any element of
the sample space is between 0 and 1(for E € S, 0 < P(E) < 1,
or P(E) =0,
or P(E) = 1),
(ii) P(S) = 1, P(not S) = 0,
(iii) For any countable collection of
disjoint sets A1,
A2, ..., all in the sample space, the probability of their union
is the sum of the probabilities (i.e. P(A1 or A2 or ...) =
P(A1) + P(A2) + ...).
A random variable is a function that maps elements of the sample sapce
into the real line. For instance, if the random experiment is to
throw
two dice, the sample space is made out of the collection of 36 duplets
(1,1), (1,2), (1,3) ..., (6,5), and (6,6). We can define the random
variable Y to be the sum of the numbers in the duplet. Then Y can
take values from 2 to 12. Since we also have an idea of the
probability for each of the duplets, we can also obtain the probability
of Y obtaining a particular value.
3
|
2/36
|
4
|
2/36
|
5
|
4/36
|
6
|
6/36
|
7
|
6/36
|
8
|
6/36
|
9
|
4/36
|
10
|
4/36
|
11
|
2/36
|
121
|
1/36
|