Probability
and its axioms
There are various approaches to probability
theory that have been used. These approaches are:
-
classical
-
relative frequency
-
subjective
-
axiomatic
The axiomatic approach is consistent
with the others and allows for the application of powerful mathematical
tools to probability theory. Here presented are the elements of the
axiomatic approach.
Kolmogorov introduced the axiomatic approach
to probability as an improvement over and generalization of previous approaches.
The Axiomatic Approach to Probability
An experiment is a process which
can be assigned a number (possibly infinite) of outcomes.
A sample space is the set of all
possible outcomes of a given experiment.
In some of the definitions below the term
"measurable" is used. Its definition is beyond the scope of this
course, but it is needed for certain infinite sample spaces to insure that
probabilities are meaningful. In the case of finite sample spaces
all sets and functions may be taken to be measurable.
An event is a measurable subset
of a sample space.
Two events are disjoint if their
intersection is the empty set.
Axioms of Probability
For any event A, a subset of the sample
space S, we assign a number P(A), called the probability of the event A.
This number satisfies the following three axioms of probability.
1. P(A) >= 0
2. P(S) = 1
3. If A and B are disjoint events,
P(A or B) = P(A)+P(B)
(Note that (3) states that if A and B are
mutually
exclusive (M.E.) (meaning P(A and B) = 0) events, the probability
of their union is the sum of their probabilities.)
There is a fourth axiom needed when dealing
with infinite families of events.
If {Ai} is a family of events
satisfying P(Ai and Aj) = 0 for i not equal to j,
then:
4. The probabilty of the union of
the Ai is equal to the sum of the probabilities P(Ai)
(The union and sum in (4) will be infinite
if the family {Ai} is infinite.)
Note that by axiom 2 no probability can
ever exceed 1 since no event can be a proper superset of the sample space.
Random variables
An intuitive introduction to random variables
Random variables and their distributions
may informally be thought of as described below. Formal definitions
will be given later.
Random Variable
-
a variable (typically represented by
x) that has a single numerical value, determined by chance, for each outcome
of a procedure
Probability Distribution
-
a graph, table, or formula that gives
the probability for each value of the random variable
Discrete random variable
-
has either a finite number of values
or countable number of values, where ‘countable’ refers to the fact that
there might be infinitely many values, but they result from a counting
process.
Continuous random variable
-
has infinitely many values, and those
values can be associated with measurements on a continuous scale with no
gaps or interruptions.
Requirements for Probability Distributions
-
S P(x)
= 1 where x assumes all possible values
-
0 < P(x) < 1 for
every value of x
Mean, Variance and Standard Deviation
of a Probability Distribution
-
Mean:
µ = S
[x • P(x)]
-
Variance:
s 2
= S
[(x - µ)2 • P(x)]
-
Variance:
s 2
= [S
x 2 • P(x)] - µ 2 (shortcut)
The standard deviation s
is the square root of the variance.
The mean, or expected value, E of a random
variable is equal to the mean of its probability distribution.
For a discrete random variable:
Example:
If a gambling game costs $1 to play and
the winning payoff is $500, where the probability of winning is 1/1000
or .001, the net gain for a loss is -$1
and for a win $499.
-
This gives rise to a random variable with
values -1 and 499.
-
The corresponding probabilities are .999 and
.001.
| Event |
x |
P(x)
|
x • P(x) |
| Win |
499
|
.001
|
0.499
|
| Lose |
-1
|
.999
|
-0.999
|
| Total |
|
|
-0.50
|
The mean = the expected value = E = -0.50,
so:
-
The expected winnings are -$0.50.
-
The expected loss is $0.50.
Formal Definitions
A random variable is a measurable
function from a sample space to the set of real numbers.
A random variable has a probability distribution
which may be specified by either of two related functions:
-
its cumulative distribution function (cdf)
or
-
its probability density function (pdf).
The cumulative distribution function
(cdf) F of a random variable X is defined as follows:
F(t) = P(X <= t)
So the value of the cdf at a number t is
the probability the random variable has a value less than or equal to t.
The probability density function
(pdf) f of a random variable X is defined as follows:
-
If X is discrete:
-
If X is continuous:
-
f(t) = F'(t)
-
F' is the derivative, or rate of change, of
F
So in the continuous case the pdf gives the
marginal cumulative probability, or the rate the cumulative probability
is increasing, at the value t.
Binomial distributions
Binomial Experiments
-
The experiment must have a fixed number of
trials.
-
The trials must be independent. (The
outcome of any individual trial doesn’t affect the probabilities in the
other trials.)
-
Each trial must have all outcomes classified
into two categories (called success and failure).
-
The probabilities must remain constant for
each trial.
Notation for Binomial Probability Distributions
-
n = the fixed number of trials
-
x = specific number of successes in
n trials
-
p = probability of success in one of
n trials
-
q = probability of failure in one of
n trials (q = 1 - p )
-
P(x) = probability of getting exactly x successes
among n trials
Binomial probability formula (the
pdf of a binomially distributed random variable)
Example
This is a binomial experiment where:
-
n = 5
-
x = 3
-
p = 0.90
-
q = 0.10
Using the binomial probability formula to
solve:
-
P(3) = 5C3
• 0.93 • 0.12 = 0.0729
Tables, calculators and computer software
are also useful for solving binomial probability problems.
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