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Lecture 8

# STAT 101 Lecture Notes - Lecture 8: Conditional Probability, Main Diagonal, Sample SpacePremium

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Richard WatermanLecture

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**preview**shows pages 1-2. to view the full**6 pages of the document.**Stat 101 - Introduction to Business Statistics - Lecture 8: Continuing Probability

Marginal Probability

● In some problems, the object of interest is the chance of two events happening

simultaneously. Another way of articulating this statement is that we are interested in

finding the association between two variables.

● In the probability tables below we have probability models for the daily returns on two

stocks, IBM and Amazon.

● If we make the assumption that the future looks like the past then we can use these

models to learn about future returns.

● These distributions are called marginal probability distributions for each variable (not

telling you about how the two stocks move together)

● The main thing to notice from these 2 tables is that AMZN is much more likely to have an

extreme return than is IBM. AMZN is described as more volatile, and this attribute can be

identified directly from the probability distribution of returns.

● If one were to hold a portfolio of these 2 stocks then the behavior of the portfolio

depends on how the 2 stocks are related. This is a key concept used in optimal portfolio

construction in Finance and we will study its measurement shortly

Joint Probability

● The probability table that describes the joint behavior of the two stock returns is called

the joint distribution.

● The table on the next slide indicates for example that there is a probability of 0.164 that

AMZN goes up by over 3%, but IBM has a return of between -1% and 1%. (Note the

highlighted cell).

● Notice that the row and column totals of the probability table are identical to the two

probability tables presented on Tables 7 and 8.

● Probabilities inside the table are the probability of that combination of returns on stock

occurring (the probabilities on the edges are the marginal probabilities

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○ There is much more chance that both stocks do well, or both do badly, rather

than one doing well and the other doing badly.

○ For example, the chances that both stocks go up by more than 3% is 0.068. The

chances that AMZN goes up by more than three percent, whereas IBM falls by

more than 3% is only 0.016.

○ It appears that the stocks move together to some extent. Again, more on this

shortly when we discuss covariance.

○ Lots of probability concentrated around leading diagonal indicating stocks are

positively correlated in their movements

● We can now ask some questions about the performance of IBM and AMZN

simultaneously.

○ Q1 What is the probability that AMZN has a return of more than 3% AND IBM has

a return of more than 3%. → 0.068.

○ Q2 What is the probability that AMZN has a return of more than 3% OR IBM has

a return of more than 3%. → 0.416.

■ for this question, you can either add highlighted cells or add probability of

AMZN obtaining more than 3%, IBM obtaining more than 3% and then

subtracting the probability of both being more than 3% (union of both)

○ Q3 What is the probability that AMZN has a return of less than -1% AND IBM has

a return of NOT less than 3%. → 0.024.

○ All these questions can be answered by returning to the joint distribution table

and identifying the events that correspond to the questions. The probabilities of

these “simple” events are then summed to find the answer

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