The expected value is the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities. Expected value is usually denoted by .
If you’re not familiar with the concept of a random variable, you can learn more about it here.
The weighted average formula for expected value is given by multiplying each possible value for the random variable by the probability that the random variable takes that value, and summing all these products. The formula can be written as:
Looking at an example will help you understand this better.
When you roll a die, you will be paidfor an odd number and for an even number. Find the expected value of money you get for one roll of the die.
The sample space of the experiment is
The table illustrates the probability distribution for a single roll of a die and the amount that will be paid for each outcome.
Using the weighted average formula, we get:
So the expected value is 1.50$. In other words, on average, you get 1.50$ per roll.
A Key Analytical Framework: Expected Value
The expected value computation provides a framework that
is extremely useful in organising thinking about data-analytic problems.
Specifically, it decomposes data-analytic thinking into (i) the structure of the problem, (ii) the elements of the analysis that can be extracted from the data, and (iii) the elements of the analysis that need to be acquired from other sources (e.g., business knowledge of subject matter experts).
As we have already said, the expected value is the weighted average of the values of the different possible outcomes, where the weight given to each value is its probability of occurrence.
For example, if the outcomes represent different possible levels of profit, an expected profit
calculation weights heavily the highly likely levels of profit, while unlikely levels of profit
are given little weight.
The same expected value formula that we wrote at the beginning of the post, can be rewritten as:
Each is a possible decision outcome; is its probability and is its business value. The probabilities often can be estimated from the data (ii), but the business values often need to be acquired from other sources (iii). Data-driven modelling may help estimate business values, but usually the values must come from external domain knowledge.
We will now illustrate the use of expected value as an analytical framework in a familiar business problem, the problem of targeting the best prospects for a charity mailing.
Targeting the Best Prospects for a Charity Mailing
Targeted marketing is a classic business problem for applying data science principles and techniques, because the fundamental structure of this problem occurs in many other problems businesses usually experience.
Fundraising organisations need to manage their limited budgets and the patience of their potential donors. Asking for donations and sending out incentive packages does come with a cost. So in any given fundraising campaign, they would like to target and approach a favorable portion of donors, those that are most likely to respond and bring the most profit to the organisation.
We would like to “engineer” an analytic solution to the problem, and our fundamental
concepts will provide the structure to do so. We are now in the Business/Data Understanding phase of the CRISP-DM cycle (if you don’t know what CRISP-DM is, you can learn more about it here).
We first have to define what exactly is the business problem we are trying to solve. At first, you might think that we want to model the probability that a customer will respond to our offer. But this approach is flawed, because different people can donate different amounts. We could have a a huge number of people respond to our offer and donate just 1$ each. But, assuming that the cost of soliciting is approximately 1$ per person, we would end up making no money. So, this is something that we have to take into account.
Let’s reconsider our approach.
Our ultimate goal in this problem is to maximise donation profit, meaning the net, after subtracting the costs of soliciting.
That means, we want to target people that are likely to respond and give us a large donation as well. We don’t only want to target people based on the probability of responding, we want to target people based on the profit that that person can get us.
And, while we do have methods of estimating probability of response, we do not have methods of directly estimating profit.
This is where expected value comes in. We can use expected value as a framework for structuring our approach to engineering a solution for our problem. Our formulation for the expected value (or cost) of targeting a specific customer x is:
where is the probability of a customer x responding, is the value we get if a customer responds, and is the value we get from no response. Since everyone either
responds or does not, our estimate of the probability of not responding is just . As we already said, we can model these probabilities by using historical data.
However, setting up the equation using the expected value framework helps us to realise some difficulties we didn’t know existed before. In this case, the value () we get from each donation varies from person to person, and we don’t know the value of the donation that each person will give until after he, or she, has been targeted.
So, let’s modify our formulation to make this explicit:
where is the value we get if a customer x responds, and is the value we get if a customer x doesn’t respond. The value of a response,, would be the consumer’s donation minus the cost of the solicitation. The value of no response,, would be zero minus the cost of solicitation.
To be complete, we also want to estimate the benefit of not targeting, and then compare the two to see whether our targeting actually has a positive effect. As you can guess, the expected benefit of not targeting is simply 0. We don’t expect people to donate randomly, without any solicitation.
Stepping back for a moment, this example illustrates why the expected value framework
is so useful for decomposing business problems. As we can see from the formula above, the expected value is a summation of products of expected probabilities and values, and data science gives us methods to estimate both probabilities and values.
Looking at historical data on people who have been targeted, we can use regression modelling to estimate how much a consumer will respond, that is, we can estimate. The same historical data can be used to estimate the probability of a customer responding to the solicitation, i.e., we can estimate.
The expected value framework allowed us to decompose a business problem into sub-problems for which we already have known techniques and solutions. We started off with a relatively complicated business problem and, using the expected value calculation, managed to decompose it into simple problems of regression and class probability estimation. The expected value framework also shows us exactly how to put the pieces back together to form the final result.
Let’s look at our formulation once again:
In this equation, we haven’t explicitly shown the solicitation (incentive) cost. The solicitation cost is the cost associated with the incentive package that we send out to each potential donor. Let’s rewrite the equation, and have c represent the solicitation (incentive) cost, and have represent the estimated donation if customer x were to respond:
Now it is up to us to decide, based on this equation, when do we want to send out an offer to a specific person. We could specify different thresholds, but the intuitive thing to do would be to mail only those people whose estimated expected donation is greater than the cost associated with mailing!
We can get that equation like this:
If we simplify it:
That is, the expected donation (left side) should be greater than the solicitation (incentive) cost (right side).
The expected value calculation allowed us to decompose a relatively complicated business problem into two simple problems of regression and class probability estimation, whose results we can recombine using the above written formula to get our final results.
The expected value calculation is extremely useful and can be applied to a variety of other business problems, some of which we will see in future posts.
The concepts for this post were taken from the book Data Science for Business, written by Foster Provost and Tom Fawcett. This book will be well worth your time, I highly recommend it.
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