welcome
to Block 5 Probability and Statistics.
My name is Lydia Montero.
We are going to see the concept of probability,
some properties about probabilities and the definition
of random variables with some structural properties
again, then let me continue.
This is the contents,
the learning objectives of the module
and introduction to the topic of probability.
Before defining probability we have to define
the concept of event and even space.
So, an experiment is the process
of collecting data from a given situation.
So, you get an observation or sample.
An event is the outcome of an experiment,
and the basic event or simple event
is just the event that cannot be decompose.
For example, tossing a die.
If we toss a die, we have to define
in this case A that is the event of observing
and odd number and B observe a number that is greater than two.
If we represent
the is these events,
we see that this is the event space notated as
omega. In the event space we have the
the representation of the simple events in this case 6.
I was proposing an ordinary die
and if I define the first, A that it is odd numbers
this is the set containing {e1, e3, e5}.
and if I define B this is obtaining
e3, e4, e5 and e6.
So, in this case we have two events A and B and the joint
event A intersect from B, in the ordinary notation
contains two simple events.
Okay, the set of all possible results
is the sample of space or the event space, depends on the book.
This is quite important.
Now, we have to
translate the concept of event
and the probability of events to formulas.
in the function, that it's the probability,
and this probability has three axioms.
The first axiom is for any event E
the probability of the event is greater or equal to zero.
The second is the probability of omega
that is the set of all possible results is equal to one.
And finally, if we have events
that are disjoint events,
this is the intersection from the set
point of view is the empty set.
Then the probability of the union of these events
is just the summatory of the individual probabilities.
Okay, For the calculation of the probabilities
is important to address the complement.
The complement of one event E is just the event
that does not contain the events included in E.
Okay, so
the probability of the E bar that is the complement
event of E is just one minus
the probability of event E,
and this takes us to the probability
of the null event, in this case, this is one minus
the probability of omega and this is zero
Joint probability.
Let's assume tossing a coin twice
the sample space could be obtaining
head and head, head and tail, tail and head, or two tails.
Okay, let's define A as obtaining two heads
and let's define B as obtaining one head.
Okay, so in this case, the joint probability is defined
as the probability of the intersection
of the two events from the point of view of sets
and this is in this case, if we have to obtain
the probability of the joint event, then
what we have to do it is to take into account the individual probabilities.
We will see that the properties. In the properties,
first of all, is a concept that is related to probability.
We have the independence of events.
The independence of events is defined
as the probability of the joint event to be equal
to the product of probabilities
is a formal statement, it is not intuitive.
Disjoint events is not the same as independent events.
Okay, we can generalize the situation to
more than two events,
in this case, the same property holds.
Let's take an example.
That is the bachelor choice.
So, we have 4000 students and 2000 women
and 2000 men and we have 2000 enrolled in engineering bachelors
and the rest in other bachelors.
So, will event A be independent from even B?
Let's use the definition.
The joint probability
is 200
divided into the total number of observations, that is 4000.
Okay, so this is 0.05.
The probability of being a woman
is 0.5 because we have 2000
observations in the table.
So, the probability of B,
studying an engineering
bachelor is, according to the table, 0.5.
Because we have 2000 students at engineering
bachelors and 4000 in total.
So the probability of the joint,
the joint probability is 0.25,
and the product of the individual
probabilities is 0.5 by 0.50, 0.25.
They are not equal.
So, these events
are not independent.
Let's focus on the additive rule.
In the additive rule, we are assuming omega
as the set of all the events in the experiment
and we have individual events A and B,
So, for any two events A
and B, the probability of the union
is the probability of A plus, the probability of B
minus the joint probability between A and B
The next concept
is very important, the conditional probability
For the conditional probability
statement, we have two events.
They belong to the same event space, and we assume that
the probability of both events is greater than zero.
At least one has to be greater than zero.
The conditional probability of B event
given A, notated as B / A
is equal to the joint probability
divided into the A probability.
So, given A independent from B,
let's see what the fact on the conditional probability.
So we have here in blue that the probability of A conditional to B
is equal to the joint probability on the numerator.
This is just the product of probability of A by probability of B
divided into the probability of B, You see that
probability of B vanishes.
So, this is equal to the probability of A event.
So, if two events are independent,
the conditional probability of one to the other
is not affected.
So, in this case, the probability of A conditional to B
is equal to the probability of A realization of event
B does not affect the realization of event A
Let's see in the case of the bachelor choice.
Let's practice a little bit.
Let's practice B engineering bachelor conditional
to A, being a woman, how to do this.
Then we apply the definition
and applying the definition
what we see is that the probability of engineering
studies conditional to women is 0.1,
and the opposite
being a student of engineering, then
A, so given a student of engineering,
the probability of being a woman, applying the definition is also 0.1.
This does not mean anything.
Nothing. It is just
the numbers.
Let's move to Bayes' rule
because this rule is extremely important in Statistics.
Bayes' rule is,
it comes from the definition of conditional probability.
So let's assume that we have a realization that is
even A, the probability greater than zero.
So, the probability of B event
conditional to A is
the probability of the join set divided into the probability of A,
but taking into account the results of the previous slide,
we can write the joint probability in the numerator as the probability
of A conditional to B, so just the reverse,
by the probability of B.
So, this process is very important because we take information
from one side to the other and then we use the information
to infer from the basic fact, the basic event.
Bayes'rule can be extended.
And this is very interesting.
Let's assume that we have a partition, a partition
of the even space.
This partition is B1, B2 ... Bk.
Okay.
Partition means that the joint event, the intersection is null.
and that they cover the whole set of all possible result that is omega.
Okay.
So based on this, we have the set of basic
results, the set of results omega.
This is divided according to the chart
in several Bi events.
So this is a partition because they recover the full rectangle
and we have A, that is an event.
So, given A,
which is the probability of any Bi?
And, this can be written
as the definition of the joint probability that we have seen that can be reverted
according to the previous slide,
and it's the probability of A conditional to Bi by the probability of Bi
and in the denominator we have the probability of A,
but for the probability of A, we have to take into account the pieces
and these pieces are the,
the joint events of B1
intersection A, B2 intersection A...
And so one.
So, this is the law of total probability.
The composition of any event A,
according to
the intersection to a given the partition of the event space.
And then random variable.
Let's move forward to the definition of random variable.
Random variable is just the numeric register
of the outcome of an experiment.
And this is practical from the mathematical point of view,
and this leads to basically two possibilities
to the discrete case with results that are integer numbers.
Not exactly.
It could be wider to set, but
the idea is this, and the continuous case.
So, the range of the values is an interval of the real numbers.
Okay.
Almost all courses deal with random variables and
this is a very convenient way to represent the outcome of a given experiment.
So, now the focus is that we have to
translate the probability that has been defined
in the events, we have to move to the values
to define the probability of the values of the random variables.
And this is going to be very easy because it's just adding up.
Okay, so we have the probability
function in the case of discrete random variables and it means that
for any value of a random variable, a discrete random variable,
the probability function ranges between zero and one,
and adding up all the probability
sorry, adding up the probabilities for all the possible values gives one.
Okay.
We have also the distribution function
that is a cumulative probability
and it's very useful.
And also it ranges between zero and one.
And is the meaning of probability,
cumulative probability.
Now to finish this tip, in
this part of the introduction to random variables and the connection
to probability, we have to define the expectation.
The expectation is the formal term for the mean.
The average in ordinary language.
So the expectation of X, capital letter,
for a given variable, random variable in the case of a discrete
random variable, it is just obtained
by adding up the values
by the probability of the values of the random variable.
And this gives the center of gravity.
So, the ordinary mean .
In the continuous case, it is a little bit more difficult because we have to use
an integral and in this interval from minus infinity to plus infinite,
we have to take into account the probability density function
that they have not defined yet. The expectation has nice properties.
As the additive property.
And just for summarizing this last slide,
we have the mean concept.
This terminology, related to random variable, is the expectation.
Usually the notation is 'mu' as expected value
for the variable X,
the computation, according to discrete random variables,
and the computation according to continuous random variable.
The variance is another statistic
that refers to the spread of the values
of the random variable around the mean.
And this can be
obtained using the summatory of the squared
discrepancies for the values with respect
to the mean, by the probability of the value.
In the case of continuous variables, we have to involve
the integrals.
And the properties, the additive property.
The expectation of the
sum of two random variables is the sum of the expectations
and the linear transformation that also holds.
If two variables are independent, the expectation of the product
is the product of the individual expectations.
This linear transformation property does not hold for variance.
In the case of variance,
if the linear transformation involves to scale
the original variable by a (real number) then
the variance of the linear transformation a X + b
is equal to 'a squared'
by the variance of the variable X.
Okay.
In the case of independency between the two variables we have
that the variance of the sum can be obtained as the sum of the variances,
and we will see some more
important concepts related to independence
in the following videos, including the 'Chi-Square' distribution
that is very important in basic statistics.
This has been everything from my site.
Thank you much for your attention.