6. Probabilities on joint events

In some problems, the object of interest is the chances of two events happening simultaneously. Another way of putting this, is that we are interested in finding the relationship between 2 variables. Most of this course has been about relationships. However, when we are talking about 2 random variables, we describe this relationship by what is called the joint probability distribution.

In the tables below we have probability models for the daily returns on two stocks, IBM and AMaZoN. If we make the big assumption that the future looks like the past then we can use these models to learn about future returns.

 

Table 3: Probability model for the return on IBM stock

Return	$< -3\%$	$-3\% - -1\%$	$-1\% - 1\%$	$1\% - 3\%$	$> 3\%$
Probability	0.056	0.212	0.392	0.232	0.108

 

Table 4: Probability model for the return on AMZN stock

Return	$< -3\%$	$-3\% - -1\%$	$-1\% - 1\%$	$1\% - 3\%$	$> 3\%$
Probability	0.288	0.132	0.108	0.096	0.376

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 is 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 Finance, and we will study its measurement in Class 10.

Today we will, see first the idea of a joint distribution.

The table below 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%.

Notice that the row and column totals of the probability table are identical to the 2 individual tables presented earlier.

 

Table 5: Joint probability model for the return on IBM and AMZN stock

	AMZN
IBM	$< -3\%$	$-3\% - -1\%$	$-1\% - 1\%$	$1\% - 3\%$	$> 3\%$	Total
$< -3\%$	0.028	0.004	0.004	0.004	0.016	0.056
$-3\% - -1\%$	0.092	0.048	0.028	0.008	0.036	0.212
$-1\% - 1\%$	0.088	0.056	0.044	0.040	0.164	0.392
$1\% - 3\%$	0.064	0.016	0.024	0.036	0.092	0.232
$> 3\%$	0.016	0.008	0.008	0.008	0.068	0.108
Total	0.288	0.132	0.108	0.096	0.376	1.000

There is much more chance that they both do well, or both do badly, rather than one doing well and the other doing badly. It appears that the stocks move together to some extent. Again, more on this next class.

We can now ask some questions about the performance of IBM and AMZN simultaneously. For example,

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

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

What is the probability that AMZN has a return of less than -1% BUT IBM has a return of more than 3%.

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.

Key probability words here AND, OR, NOT.