쉐도잉 연습: 1. Introduction to statistics - Lecture 1 - 영상으로 영어 말하기 배우기

Dear students, welcome.

My name is Ludovico Biagi, I am professor in Politecnico di Milano.

And with this slot of lectures I would provide you a general introduction to statistics

and probabilities and what it is needed to better understand more complicated ideas relevant to geospatial intelligence.

The slot of lectures on statistics will last about more or less five hours.

It is split in several slots and each one's slot concerns a specific topic

and each one's slot is accompanied by exercises that you can do by yourself.

and questions to let you understand if you have completely understood what is contained in the lectures.

First of all, I would start with a general outline about the following slides that we will see in the next slots.

We will start with the general definition of probability, the general properties of probabilities,

then we will introduce the idea of conditional probability

and this leads to the definition of the theorem of bias

and the idea of total probability and then we will move to the concept of independent events.

After the discussion of probability we will move toward the idea of random variables.

We will start in one dimension, so we will define what is a random variable,

we will define what is the distribution and the density functions of a random variable, we will see some specific examples of random variables,

because these examples are very important for the following.

We will define what is the function of one random variable, and then we will see what is the mean and the variance of a random variable.

Then we will move to random variables in two dimensions, I mean x and y.

We will define them, we will see what is the joint density function,

what are the marginal distributions of the two components of a random variable in two dimensions.

again what is the mean the covariance, the propagation laws for the mean and the covariance,

the conditional distributions and the concept of independence between the two components of a random variable in two dimensions.

The next slot will be about the generalization to Rn dimensions.

so again we will see how the properties of random variables in two dimensions can be extended to a general n-dimensional space.

We will see also the differences in Rn with respect to R2.

And finally, a slot will be devoted to the so-called asymptotic behavior of random variables

that puts a very important link between the theory of probability and the practice of observations.

I mean why and how the general theory of probabilities

and random variables can be applied in the management and analysis of observations.

Finally, we will see a particular example of estimator for random variables, that is least squares.

So we will put the general position of the problem, we will see what are for least squares the estimation principles,

what are the estimates provided by least squares, both of the unknowns and of the covariances of the estimated parameters.

Then in the last, very last part, we will see what does it mean to make an hypothesis testing

and in particular we will see how to apply this general principle to the global model

and to the individual observations in least squares.

So let's start with the probability and in the order we will see the axiomatic definition of probability,

the properties of a probability, what is the conditional probability,

what is the total probability of one event and what is the bias theorem.

We are not going to prove theorems just to enunciate and discuss them and finally we will see what are independent events.

So in general what are stochastic experiments and what is their modeling?

Let's start with the definition of a probability model, we can call it capital omega.

The elements of a probability model are called also experimental outcomes and we will denote them with the lower case omega.

The subset of capital omega are named events.

In this picture you see that in orange it is the complete set,

so capital omega, and the white area is a specific event,

capital A, named capital A.

Just to complete the model we need clearly to assign probabilities to all the events capital A

that compose the capital omega set.

So we can start with the axiomatic definition of probability.

Let's take an event capital A.

The probability of capital A is a certain number

that we can symbolize with the p of capital A that is related to capital A.

This number must satisfy the following properties axioms.

The probability of an event is not negative, so it is bigger or equal to zero.

The probability of the whole set of events, so capital omega, is equal to one.

Then let's take two exclusive events, capital A and capital B,

then the probability of their union, the probability that they happen together,

is given by the probability of A plus the probability of B.

Let's generalize the idea of the tiered axiom.

By looking to the picture we have again the capital omega set of events.

Let's suppose you have three events in this case A, B and C that are depicted with the white, blue

and red areas, the probability of A union B union C is equal to

the probability of A plus the probability of B plus the probability of C.

This can be extended to any finite number of terms

and this could be extended also for an infinite but countably many terms.

This is the axiom of infinite additivity that states that the union,

the probability of the union of an infinite number of events a1, a2, etc, etc, etc,

is simply given by the sum of probability of a1 plus

probability of a2 plus up to an infinite sum on all these set of events.

More and more about properties.

The probability of the impossible event, or if you prefer the zero event, is equal to zero.

Moreover, let's look to the picture.

You have capital omega, that is the whole set of possible events.

You have the event A, again the white area.

The complementary of A is defined as omega minus A, so all the events that are not A,

and the probability of the complementary of A is 1 minus the probability of A.

Now the probability of an event A is always between 0 and 1,

0 for the impossible event and 1 for capital omega taken as a wall.

Let's consider two events, B and A, and let's suppose that B is a subset of A.

In this case, the probability of B is smaller or equal to the probability of A.

Now let's suppose A and B, on the contrary, are two events with an intersection exactly in the picture.

You see there is an orange omega, in white A, in blue B, and you see that they have an intersection.

In this case the probability of A union B is exactly

equal to the probability of A plus the probability of B minus the probability of their intersection.

An experiment, capital omega, is specified in terms of the probabilities of all the events that compose the experiment.

However, considering the previous axioms, we don't need to assign probabilities to every event.

In particular, if capital omega consists of n outcomes,

the experiment is specified in terms of the probabilities of the elementary events lowercase omega.

The foregoing can also also, if capital omega consists of a countable number of outcomes.

On the contrary, it doesn't hold when the elements of capital omega are not countable,

for example the real valued points of an interval that is not countable.

Let's consider an experiment that can output a certain number capital N of outcomes.

These outcomes are said to be equally likely

when they have exactly the same probability and each one probability is exactly equal to 1 over capital N.

Now let's suppose that an event consists of NA outcomes.

Then the probability of A can be simply written as Na over N.

As an example let's consider the classical coin or the classical die.

We can say that the coin is fair when its outcomes are equally likely,

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that is, the probability of heads is equal to the probability of tails and they are both equal to 1 over 2.

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and let's look to a die, a die has six possible outcomes from one to six,

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they are equally likely when probability of one is equal to probability of two up to probability of six,

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and this clearly means that each one probability is one over six now let's suppose you toss a coin twice,

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so you can have four different possible outcomes, events.

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Indeed, the capital omega is composed by a head head,

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head tail, tail head and tail tail.

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If these four outcomes are equally likely, then each one of them has probability equal to 1 over 4.

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Now let's suppose that you want to check the probability of the event A that both the second toss,

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pardon, has outcome as tail.

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This event A is generated by age T or TT.

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So two possible outcomes generate the event and so the probability of the event is 1 over 2.

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Now let's suppose that our set capital omega consists not of a number countable of outcomes

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but of a non-countable number of outcomes.

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For example the possible points in the real line, in this case capital omega is defined from minus infinite to plus infinite.

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In this case the experiment can be specified not in term of a probability but in term of a density function.

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Density function alpha is named in this slide is always bigger

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or equal to zero and integral between minus infinite and plus infinite of alpha is equal to 1,

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and in particular alpha is such that the probability

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that an outcome is comprised between a minimum value a

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and the maximum value b is given by the integral from a to b of alpha in dt.

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Now we can move to the idea and the properties of the conditional probability.

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Let's take any event M such that the probability of M is bigger than zero,

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strictly bigger than zero, and let's give another event A.

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We can compute the ratio probability of A intersection M divided by probability of M.

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You see in the picture in the bottom of the slide the example you have capital you have m in red,

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you have a in gray and they have an intersection.

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The ratio we have just defined is called a conditional probability of a assuming

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that m happened and it is denoted by the symbol probability of a given m

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and okay and it is the ratio of p intersection m divided by m.

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The meaning of conditional probability can be simplified as follows.

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You know that an event m happened,

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you just ask what is the probability that given m, a happens.

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The properties of the conditional probability are the following.

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Let's take again an event M

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and as a range as a ranges over all the events of capital omega we have

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that the probability of A given M is always bigger

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or equal to zero the probability of omega given M is equal to 1,

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and in particular, they take two different events, A and B.

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In case the intersection between A and B is void, we can write

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that the probability of A union B given M is the

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probability of A given m plus the probability of b given m.

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For example, let's take again the example of a die.

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We remember that the possible outcomes of a die are six, exactly from one to six.

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we want to determine the probability to obtain the outcome 2 assuming that an even outcome is given.

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So the probability of M, probability of even, is 1 over 2.

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The probability of 2 given even is the probability of A intersection M divided by the probability of M.

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The probability of A intersection M is 1 over 6, the probability of M is 3 over 6,

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so the probability of A given M is 1 over 3.

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In the picture in the bottom of the slide you see exactly the example, capital omega,

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m that is the set of even outcomes 2 4

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and 6 and a that is the only one outcome equal to 2.

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Now let's move to the concept of total probability

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and the theorem of bias that is very important for many many many applications.

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Now let's suppose to have a partition of capital omega.

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I mean that we have a set of events a1, a2, up to a m,

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such that the intersection between ae and aj is zero for each couple of ej,

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and the union of all the events a is the whole capital omega.

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Now let's suppose that you have an event B.

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The probability of B can be written as the probability of B given A1

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that multiplies the probability of A1 plus the probability of B

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given A2 multiplied by the probability of A2 summed up to the whole set of events A,

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so summed up to AM.

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So let's see the idea with a simple example.

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We have a political poll and we have recorded the following results.

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Among all the voters, 70% are male and 30% are female.

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Among males, 40% are Republican and 60% are Democrats.

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Among families 45% are Republican and 55% are Democrats.

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You need to find the probability that a voter selected at random is Republican.

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So from a graphical point of view, you see in the picture the set capital omega all the

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votes the orange set is split by a line into male

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and female voters and the outcomes are split in a set

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that is again orange of republicans and a red set of Democrats that are clearly shared between families and males.

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So the probability of a male voter is 0.7,

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the probability of a family voter is 0.3, the probability of a Republican given

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m is 0.4 the probability of R given F is 0.45

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so the probability of R is the probability of R given

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M multiplied by the probability of M plus the probability of R given F multiplied by the probability of F.

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If you make the computation, you find that it is equal to 0.415, that is 41.5%.

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Now a very important theorem that we are not going to prove

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but just to discuss let's suppose that you have a partition of capital omega

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and we call this partition again a1 a2 up to a m it means

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that the their intersection is void for each couple

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and their union is the wall omega now let's suppose

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that you have another event b

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that in the picture is depicted as the red area

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and the probability of a given a given b so probability of ae given b

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is equal to the probability of b given ae

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that multiplies the probability of ae divided by the sum of probability of b given ae

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that multiplies probability of ae summed on all the a events,

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so summed on all the m a events.

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And with the help of two coins we can depict an example.

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Let's suppose we have two coins a and b.

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Coin a is fair, so the probability of head is equal to the probability of tail

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while coin of b is loaded with a probability for s

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that it is 2 over 3

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so probability of tail for b is 1 over 3 let's

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suppose you pick up one of the coin at random without knowing

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if it is a or b let's suppose you toss it

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and the outcome is s what is the probability that we picked the fair coin.

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So the probability can be computed as probability of F given add,

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so probability of add given A

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that multiplies probability of A over the probability of edge given A

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that multiplies probability of A plus probability of edge given B multiplied by probability of b.

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You make all the computations and you find that the probability is 3 over 7.

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Now let's move to the idea of independent events.

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We have two events, a and b.

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They are called statistically independent when the probability of their intersection is exactly equal to the product of their individual probabilities.

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So from a mathematical point of view we can write

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that probability of A intersection B is equal to probability of A that multiplies probability of B.

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What happens when two events are statistically independent for conditional probabilities?

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okay the probability of A given B

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that is defined as the probability of A intersection B divided

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by probability of B is equal to probability of A

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that multiplies probability of B divided by probability of B

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that is clearly equal to probability of A in the same way we see

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that probability of B given A is simply given by probability of B.

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Again let's help with a simple example.

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Let's suppose we have a fair coin and a fair die and let's suppose we toss both of them.

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Clearly the two experiments are independent.

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So let's suppose we want to compute the probability

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that 5 shows on the die and the output of coin is 8

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so the probability of the

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that the two outcomes happen together is the probability of the intersection

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that in this case because of the independence is the product of the individual probabilities

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and so we have 1 over 12 so this ends

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this first set of slides about statistics and at the end we find questions

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that allow you to self-evaluate yourselves i mean

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if you are able to answer to the questions you have

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understood the ideas on the contrary case it is better you try to look again to the lecture

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so the first question is what is a stochastic experiment outcome

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and what is an event the second question is what is

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the set capital omega of a probability space the third question

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is what is the set how it is defined the set of all the possible events

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and what are the properties of this set now you should be able to give the axiomatic definition of probability

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and now let assume

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that the set omega is discrete is it sufficient to assign

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the probability to each outcome to define it why it is the answer give an example of uniform distribution of probability

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possibly not the coin or the die, if the set capital omega is continuous,

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how we can specify the probability of an event,

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give the definition of conditional probability and the definition of of stochastically independent events,

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what is a partition of capital omega, and discuss the bias theorem.

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Moreover, in the exercises folder of this set of lectures,

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you will find a few very simple exercises that you should try to solve by yourself.

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Thank you.

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