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So what have we learned in the second week of this MOOC course?

Well, here we've been concerned with trying to quantify uncertainty with

probability.

Now do appreciate that probability theory is a vast topic and

we could spend a year or more looking at this in intricate detail.

But nonetheless, some of the basic principles of probability have been shown.

For example, we've looked at deriving some simple probability distributions.

Recognizing that if we consider all possible outcomes of an experiment,

how we can attach probabilities to represent the likelihood of each of those

different outcomes.

We've also considered how we actually can quantify probability, and

there are different methods for doing this.

Sometimes we may just have to resort to subjective estimates.

What's the probability that a world war breaks out next year?

Well, clearly we can't conduct experiments to

a sort of at a relative frequency approach to this.

It simply comes down to subjective estimates based on what we read in

the media and our knowledge about the world.

But sometimes we have got to just go down the subjective estimate route because

there is no real viable alternative.

The relative frequency approach, conduct an experiment a large number of times and

see what proportion of the times a particular outcome occurs.

And that could represent numerically the probability of that particular outcome.

And thirdly, we also consider some theoretical derivations.

For example, if we had a fair die,

we recognize there are six equally likely outcomes.

And theoretically,

we would attach a one over six probability to each of those values occurring.

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Now having considered some simple distributions,

we then introduce the very important concept of expectation whereby we were

focusing on one key attribute of those distributions.

Namely on average, in the long run,

what value of this variable would we expect to get?

So for example, in the case of rolling a fair die,

there we found that the expected score was 3.5.

Recognizing that we would never get 3.5 on a single roll of the die, but

if we imagined performing a die experiment a very large number of times,

then on average that would be approximately the value we would get.

We also considered Bayesian updating.

How we can update our beliefs probabilistically as we looked at it,

in light of new information.

For example, rolling a die and if you're told that an even or

an odd number has occurred, this will allow you to update your assessment of how

likely you are to have rolled a six or indeed any other particular value.

Indeed, we also saw some links to the Monty Hall problem we looked at,

at the very start of week one.

Whereby when that door B was revealed to have a goat behind it,

this would allow us to update our beliefs probabilistically about how

likely the sports car was to be behind any of those doors.

And then we round it off with a look at parameters and

just a brief introduction of a few probability distributions.

Namely, the Bernoulli, the binomial and the Poisson distributions.

But as I said, there are hundreds,

arguably thousands of different distributions out there.

And clearly it would take a lot of time to be acquainted with all of them.

But nonetheless, week two has been very important to us, thinking ahead to some of

these statistical inference that we wish to do later on in this course.

Because all of these statistical methods that you will see within this MOOC

will be built on some of the sort of probability foundations that we've

seen within this week.

So perhaps one of our key takeaways is to consider that

statistics and probability, as a discipline or

disciplines really is a cumulative type of study.

Namely things you learn very early on in the course are required to understand

later material.

So you will see some of the terms we've

already discussed appear again in later material.

So this is the end of week two and we're will we be headed in week three?

Well this is where we start to introduce some simple descriptive statistics.

Whereby we're trying to describe the world in a statistical way.

So join me for week three and we'll look through at some of that material.

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