Showing posts with label Bivariate Probabilities. Show all posts
Showing posts with label Bivariate Probabilities. Show all posts

Monday, July 14, 2014

Bivariate Probabilities

In this section we introduce a class of problem that involve two distinct sets of events, which label A1, A2,…..An and B1, B2,…………,Bk. These problems have broad application in business and economics. They can be studied by constructing two-way tables that develop intuition for problem solutions. The events Ai and Bj are mutually exclusive and collectively exhaustive within their sets, but intersections  can occur between all events from the two sets. These intersections can be regarded as basic outcomes of a random experiment. Two sets of events, considered jointly in this way, are called bivariate, and the probabilities are called bivariate probabilities.

The following Table illustrates the outcomes of bivariate events labeled A1, A2,…..An and B1, B2,…………,Bk. If probabilities can be attached to all intersections  , then the whole probability structure of the random experiment is known, and other probabilities of interest can be calculated.

As a discussion example, consider a potential advertiser who wants to know both income and other relevant characteristics of the audience for a particular television show. Families may be categorized, using Ai, as to whether they regularly, occasionally, or never watch a particular series. In addition, they can be categorized, using Bj, according to low, middle, and high-income subgroups. Then the nine possible cross-classifications can be set out in the form of the above Table, with i = 3 and k = 3. The sub setting of the  population can also be displayed using a tree diagram, as shown in the following Figure. Beginning at the left, we have the entire population of families. This population is separated into three branches, depending on their television viewing frequency. In turn, each of these branches is separated into three sub-branches, according to all combinations of viewing frequency and income level. As a result, there are nine sub-branches corresponding to all combinations of viewing frequency and income level.  




Figure 1: Tree diagram for television viewing and income example

Now it is necessary to obtain the probabilities for each of the event intersections. These probabilities, as obtained from viewer surveys, are all presented in Table 1. For example, 10% of the families have high incomes and occasionally watch the series. These probabilities are developed using the relative frequency concept of probability, assuming that the survey is sufficiently large so that proportions can be approximated as probabilities. On this basis, the probability that a family chosen at random from the population has a high income and occasionally watches the show is 0.10.



Table 2: Probability for Television Viewing and Income Example.

Viewing Frequency
High Income
Middle Income
Low Income
Totals
Regular
0.04
0.13
0.04
0.21
Occasional
0.10
0.11
0.06
0.27
Never
0.13
0.17
0.22
0.52
Totals
0.27
0.41
0.32
1.00

Joint and Marginal Probabilities

In the context of bivariate probabilities the intersection probabilities,  are called joint probabilities. The probabilities for individual events, P(Ai) or P(Bj), are called marginal probabilities. Marginal probabilities are at the margin of a table and can be computed by summing the corresponding row or column.
To obtain the marginal probabilities for an event, we merely sum the corresponding mutually exclusive joint probabilities:
Continuing with the example, define the television watching subgroups as A1, “Regular”; A2, “Occasional”; and A3, “Never.” Similarly define the income subgroups as B1, “High”; B2, “Middle”; and B3, “Low.” Then the probability that a family is an occasional viewer is


Similarly, we can add the other rows in Table above to obtain P(A1) = 0.21 and P(A3) = 0.52.

We can also add the columns in Table to obtain

             P(B1) = 0.27                     P(B2) = 0.41                   P(B3) = 0.32

Marginal probabilities can also be obtained from tree diagrams such as Figure 2, which has the same branches as Figure 1. The right-hand-side contains all of the joint probabilities, and the marginal probabilities for the three viewing frequency events are entered on the main branches by adding the probabilities on the corresponding sub-branches. The marginal probabilities for the various events sum to 1 because those events are mutually exclusive and mutually exhaustive.



Figure 2: Tree diagram for the television viewing-income example, showing
               joint and marginal probabilities.                                                      

In many applications we find that the conditional probabilities are of more interest than a the marginal probabilities. An advertiser may be more concerned about the probability that a high-income family is watching than the probability of any family watching. The conditional probability can be obtained easily from the table because we have all the joint probabilities and the marginal probabilities. For example, the probability of a high-income family regularly watching the show is



Table 3 shows the probability of the viewer groups conditional on income levels. Note that the conditional probabilities with respect to a particular income group always add up to 1, as seen for the three columns in Table 3. This will always be the case, as seen by the following:

The conditional probabilities for the income groups, given viewing frequencies, can also be computed , as shown in Table 4, using the definition for conditional probability and the joint and marginal probabilities.

To obtain the conditional probabilities with respect to income groups in Table 2, we divide each of the joint probabilities in a row by the marginal probability in the right-hand column.

Table 3: Conditional Probabilities of Viewing Frequencies , Given income levels.


Viewing Frequency
High Income
Middle Income
Low Income
Regular
0.15
0.32
0.12
Occasional
0.37
0.27
0.19
Never
0.48
0.41
0.69
Totals
1.0
1.0
1.0

Table 4: Conditional Probabilities of Income Levels, Given the Viewing Frequencies.


Viewing Frequency
High Income
Middle Income
Low Income
Totals
Regular
0.19
0.62
0.19
1.0
Occasional
0.37
0.41
0.22
1.0
Never
0.25
0.33
0.42
1.0

We can also check, by using a two-way table, whether or not paired events are statistically independent. Events A and B are independent if and only if their joint probability is the product of their marginal probabilities—that is, if









Odds

Odds are used to communicate probability information in some situations. For example, a sports analyst might report that the odds in favor of team A winning over team B are 2 to 1. Odds can be converted directly to probabilities, and probabilities can be converted to odds using the following equation.

The odds in favor of a particular event are given by the ratio of the probability of the event divided by the probability of its complement. The odds in favor of A are










Similarly, if the odds in favor of winning are 3 to 2, then the probability of winning is 0.60. Note that 0.60/0.40 is equal to 3/2.

Overinvolvement Ratios

There are a number of situations where it is difficult to obtain desired conditional probabilities, but alternative conditional probabilities are available. It may be difficult to obtain probabilities because the costs of enumeration are high or because critical, ethical, or legal restriction prevents direct collection of probabilities. In some of those cases it may be possible to use basic probability relationships to derive desired probabilities from available probabilities.

Suppose that we know 60% of the purchasers of our product have seen our advertisement, but only 30% of the nonpurchasers have seen the advertisement. The ratio of 60% to 30% is the overinvolvement of the event “Seen our advertisement” in the purchasers group, compared to the nonpurchasers group. In the analysis to follow we show how an overinvolvment ratio greater than 1.0 provides evidence that, for example, advertising influences purchase behaviour.

The probability of event A1, conditional on event B1, divided by the probability of A1, conditional on event B2, is defined as the overinvolvement ratio











Consider a company that wishes to determine the effectiveness of a new advertisement. An experiment is conducted in which the advertisement is shown to one customer group and not to another, followed by observation of the purchase behaviour of both groups. Studies of this have high probability of error; they can be biased because people who are watched closely often behave differently than they do when not being observed. It is possible, however, to measure the percentage of buyers who have seen an ad and to measure the percentage of nonbuyers who have seen the ad. Let us consider how those study data can be analysed to determine the effectiveness of the new advertisement.

Advertising effectiveness is determined using the following analysis. The population is divided into

                   B1 : Buyers
                   B2 : Nonbuyers

And into
                    A1 : Those who have seen the advertisement
                    A2 : Those who have not seen the advertisement.





Similarly, we can define the conditional odds, in which we use the ratio of the probabilities that are both conditional on the same event. For this problem the odds of a buyer conditional on “Have seen an advertisement” are
If the conditional odds are greater than the unconditional odds, the conditioning event is said to have influence on the vent of interest. Thus, advertising would be considered effective if


















This result shows that, if a larger percent of buyers have seen the advertisement, compared to nonbuyers, then the odds in favor of purchasing conditional on having seen the advertisement are greater than the unconditional odds. Therefore, we have evidence that the advertising is associated with an increased probability of purchase.