Teaching Course Modules Involving Multiple Parameters
By Russell Jay Hendel
Expanding Horizons, August 2025
Hypothesis testing is a good illustrative example of a course module with many parameters.[1] How do you teach such a module? A naive approach might use the following rule of thumb: “If the sample size is large, use a normal distribution; if it is small, use a t-distribution.” But this is inaccurate for several reasons: If the population is assumed to be normal and the population variance is known, then the normal distribution is exact even for a small sample. Additionally, if the sample size is small but the population is not assumed to be normal, a t-distribution may lead to misleading results, especially if the population is skewed. In such a case, nonparametric methods are typically the best.
This illustrates the characteristics of the problem—the module involves many parameters. Attempting to provide a rule with just one parameter (the size of the sample) is not adequate. Similarly, an attempt to be completely descriptive overwhelms the student, which inhibits learning.
It is natural at this point to review the literature to see what others are doing, but there is considerable confusion among available sources, including university course webpages, textbooks and software. A survey of a variety of websites shows that some approaches omit complete details; others give fancy flowcharts that are not helpful to students learning the material; and still others use a stratified approach, starting with rules of thumb and gradually building up, but this too has drawbacks since students will have multiple versions of rules in their notes and may lack full clarity on the final approach. One paper that surveyed many textbooks showed similar confusion among the approaches, and points out that supportive software for these courses often suffer from the same problem.[2]
A Supportive Technique from Mathematical Logic and Electrical Engineering
Most readers are familiar with the disjunctive normal form (DNF) from mathematical logic. The DNF is a standardized method to describe Boolean rules dependent on multiple parameters. A sister normal form, the prime implicant normal form (PINF) accomplishes the same thing but with fewer symbols and is ideal for solving the pedagogical problem already outlined. Table 1 presents the PINF for hypothesis testing, this article’s introductory example of multi-parameter course modules. Other examples will be presented later.
Table 1
Tabular Presentation of PINF for Determining a Method for Hypothesis Testing
| Issues or Parameters to Be Considered | Method to Be Used | ||
|---|---|---|---|
| Is the sample size large? | Is the population assumed to be normal? | Is the population variance known? | |
| Yes | Normal distribution | ||
| Yes | Yes | Normal distribution | |
| No | Yes | t-distribution | |
| No | No | Non-parametric methods | |
To appreciate the PINF, Table 2 presents the DNF for the first row of Table 1. Technical definitions of PINF and DNF can readily be found in both introductory logic and electrical engineering textbooks; this example suffices with some heuristics.
The DNF requires filling in all columns in each row. The entire DNF table must present all possible column combinations of Yes and No. Contrastively, the PINF allows blank cells that are interpreted to mean “whether Yes or No.” For example, the second row of Table 1 is interpreted to mean that a normal approximation should be used if both the population is normal and the population variance is known independent of whether the sample size is large or small. The rule for filling out rows in the PINF is as follows: “Fill in the minimal number of columns. If leaving a column cell empty would lead to a false result, then that column is part of the minimal number of columns.” To illustrate this principle, if in row 2 of Table 1 we changed the Yes in the second column to an empty cell, the row would incorrectly state that a normal distribution is used if the population variance is known even if the sample size is small and the population is not assumed to be normal. This is not true; hence, the two columns in the second row must be filled in as shown. Textbooks explain the mechanical methods for arriving at a PINF.
Table 2
Tabular Presentation of DNF Corresponding to the First Row of Table 1
| Issues or Parameters to Be Considered | Method to Be Used | ||
|---|---|---|---|
| Is the sample size large? | Is the population assumed to be normal? | Is the population variance known? | |
| Yes | Yes | Yes | Normal approximation |
| Yes | Yes | No | Normal approximation |
| Yes | No | Yes | Normal approximation |
| Yes | No | No | Normal approximation |
Before continuing, note that we have abstained from making technical statistical remarks. For example, we do not opine on what a large sample is. Most textbooks use 30 as a minimum number for large sample size, but this is only valid if the significance level is 5%. For smaller significance levels, 30 would not be adequate.
Advantages of the PINF Method
The PINF approach has several benefits:
- It is complete and accurate.
- It is brief and compact, so it is not overwhelming for the student.
- It allows simultaneous perception of accurate rules of thumb and complete rules. (For example, a one-parameter rule of thumb implicit in Table 1 is “For a large sample, use the normal distribution.”)
One important use of the PINF approach is in remediation. Student errors in multi-parameter modules often arise from oversimplifying Boolean rules. The PINF table allows an instructor to walk through the underlying cognitive fallacies that create such errors. This can be very useful for intermediate students who need a little push to succeed in the course.
Actuarial Examples
Having presented the basic approach advocated by this paper for multi-dimensional course modules, we proceed to apply this method to a variety of modules in the preliminary exam actuarial courses. Actuarial courses are typically described as “hard,” “technical” and “difficult to master.” A preferred description would be “actuarial courses have many multiple-parameter methods.” In other words, the driver of course difficulty is the multi-dimensionality of the course modules.
We will call the hypothesis-testing example presented earlier Example 1, with those in this section numbered sequentially. Each example has a pedagogical point and uniqueness. The point of Example 1, hypothesis testing, as already indicated, is that multi-parameter modules present challenges to successful instruction because of the simultaneous need to maximize student mastery and avoid statements that can mislead students. The PINF seems a good approach to meet these needs.
Example 2—Sums of Independent Random Variables
The point of Example 2, sums of independent random variables, is that even when the number of parameters is small, challenges can arise if the parameters are not two-valued (yes or no). Table 3 presents the associated PINF.
Table 3
PINF for Combining Independent Identically Distributed Random Variables
| Issues for Combining Independent Identical Distributions | The Combined Distribution | |
|---|---|---|
| Distributions combined | Are there restrictions on the parameters (besides independence)? | Is the combined distribution identically distributed as the distributions combined? |
| Normal | No | Yes |
| Poisson | No | Yes |
| Binomial | Yes (A common trial probability is required.) | Yes |
| Gamma | Yes (A common scale factor is required.) | Yes |
| Bernoulli | Yes (A common trial probability is required.) | No (The combined distribution is binomial.) |
| Exponential | Yes (A common parameter is required.) | No (The combined distribution is gamma.) |
The table should be self-explanatory. Because the cell values are not binary (Yes or No), there is an added richness in the underlying theory despite the small number of columns. For example, sums of independent Poisson variables are Poisson distributed even if the combining distributions have different parameters; contrastively, sums of binomial variables are only binomial if they share a common probability.
Example 3—Accumulation Functions
The point of Example 3 is that even if the number of parameters is minimal, challenges in both instruction and learning can arise if certain cases are rarer exceptions. Table 4 presents the corresponding PINF.
Table 4
PINF for the Module Accumulation Functions
| Money Growth Method | Does relative time (i.e., distance from begin to end time) suffice? |
|---|---|
| Compound and nominal interest and discount | Yes |
| Simple interest and discount | Yes |
| Variable force | No |
| Polynomial and rational functions, growth functions | No |
As indicated, the rule depends on a minimal number of parameters (in fact, one parameter). Many textbooks simply give the eight accumulation functions without emphasizing, at the beginning, the subtle distinction that relative time suffices for many functions whereas absolute time is needed for money growth under variable force, polynomial and rational functions. As indicated earlier, such PINF tables are useful for remediation for intermediate students who benefit from the explicit identification of pitfalls.
Example 4—Annuity Functions
The novelty of this example is that what is being addressed—the cases—are the number of distinct formulas needed to solve problems. Here, we echo the concerns of Example 1, hypothesis testing, that rules of thumb may be too simplistic and potentially mislead the student, even if the rules of thumb are refined later on. The typical rule of thumb found in most textbooks is that annuities are either level, arithmetically increasing/decreasing, or geometrically increasing/decreasing. This could (and in fact often does) mislead students into thinking that three formulas are needed. In actuality, seven formulas are needed, as shown in Table 5 which presents the PINF.
Table 5
PINF for Seven Annuity Types and Their Formulas Sufficing for All Annuity Problems
| Annuity Type | Formula |
|---|---|
| Level (finite) | (1 – vn) / i |
| Level perpetuity | 1 / i |
| Purely increasing annuity | 1 / i * ((1 – vn) / d – nvn) |
| Arithmetic annuity (increasing or decreasing finite or infinite)* | (f – d) (1 – vn) / i + d / i ((1 – vn) / d – nvn) |
| Geometrically increasing or decreasing annuity (finite-due) | (1 – vn) / d′, with d′ the discount factor for i′ the inflation-adjusted interest rate |
| Geometric perpetuity (immediate) | 1 / (i – g), with i the interest rate and g the inflation rate |
| Nested† | Product of (i) the consecutive annuity at the interest rate for the consecutive period and (ii) the nonconsecutive annuity (due) at the interest rate for the nonconsecutive periods |
*In the formula, f stands for the first payment, and d stands for the common difference of payments (whether positive or negative). As noted, the formula works for both increasing and decreasing payments as well as for finite and infinite payments.
†A typical nested annuity might be level monthly, increasing yearly. In this case, the monthly payments are consecutive monthly payments. Contrastively, the increasing payments are annual and nonconsecutive. Since this method calculates the nested annuity exclusively using present values, the nonconsecutive annuity will be treated as due.
Several subtleties are connected with this example. First, there is no unique correct answer. We have selected a single formula for arithmetic annuities that subsumes arithmetic annuities that are both increasing and decreasing as well as finite and infinite. However, it would be perfectly legitimate to present multiple formulas (e.g., the formula for increasing annuities and decreasing annuities), leading to a different table. Also note that the single formula assumes knowledge of the simpler case of purely increasing annuities.
The second subtlety addresses the distinctness of enumeration. A candidate fluent in limits can instantly derive the formula for perpetuities (whether of level or increasing annuities) from the corresponding formulas for finite annuities. However, candidates not fluent in limits may need two formulas.
The third subtlety addresses what we have called “nested annuities” in Table 5. By this we mean, for example, an annuity that is level monthly and increasing (say, geometrically) yearly. Textbooks, while covering this annuity, do not identify it with a name, nor do they indicate that it has its own formulas that relate to other annuity formulas. A major point of a PINF table is coverage of all distinct cases.
These subtleties highlight the focus of the PINF. The focus is not on a single right answer, but rather on a simultaneously compact and comprehensive form for presenting the answer.
Example 5—Equivalence Principle
The point of this example is to address many multiple parameters and, in fact, a nonfixed number of parameters (in the example cited, we have five parameters but discuss the possible addition of a sixth). Here already, the textbooks are aware of the problem of the simultaneous need to be compact and comprehensive, using heuristics to assist the student. For example, premium calculations begin with the heuristic formula:
(1) Expected present value of inflows = Expected present values of outflows
A further heuristic is given by the following formula:
(2) Outflows = Expenses + Insurance payments
It is tempting to try to be exhaustive, for example, by using the following equation:
(3) Expenses = Flat expenses + Uniquely initial year expenses + Shared initial and renewal years expenses
where the terms “uniquely initial year” and “shared initial and renewal years” can be illustrated by a 10% of premium initial year expense with 1% premium renewal expenses; here, 9% of premium is uniquely initial, while 1% premium is shared in both the initial and the renewal years.
To illustrate the nonfixed number of parameters, consider an insurance product with a return of premium option. The candidate can immediately deal with the new situation by classifying return of premiums as an outflow. Thus equation (2) is modified to include return of premiums.
In reality, it might be better to consider equations (1) through (3) a hierarchical approach distinct from the PINF approach. We have brought this example because first, it has commonality with the other examples, all of which deal with multi-parameter rules; second, the problem of simultaneous compactness and completeness is addressed through a hierarchy of rules; and third, the hierarchical approach allows plasticity and the introduction of additional parameters. For these reasons, we suffice with equations (1) through (3) and our remarks about return of premium.
Conclusion
This paper has dealt with the instruction of course modules characterized by multiple parameters, introducing two methods: the hierarchical method and the PINF method. It shows that the PINF method is simultaneously compact and complete and even allows partial rules of thumb. We have explored several subtleties in the PINF method, including its utility for remediation when the number of parameters is small (Example 2), its elasticity when the parameters are not strictly binary (Example 3), and its ability to avoid the pitfalls of rules of thumb (Examples 1 and 4). These techniques are easily mastered, immediately applicable and helpful in instruction.
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Russell Jay Hendel, Ph.D., ASA, participates in the Actuarial Science and Risk Program at Towson University, a Center for Actuarial Excellence. He is currently vice chair of the Leadership and Development council. Russell can be reached at RHendel@Towson.Edu.