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Bayes’ Theorem is a mathematical formula that deals with conditional probabilities. That is, it provides a means for updating probabilities based on relevant evidence that has occurred. Also, the theorem is useful when trying to solve a probability problem that seems intractable if not unsolvable at first pass. For example, suppose you find yourself standing outside of three darkened rooms labeled A, B, and C, and you know that in each room there are two people, either male (MM), female (FF), or one of each (MF). Therefore, P(M|A)=1.0; P(M|B)=0.5 and P(M|C)=0. See the following diagram: You take a flashlight and enter a room at random where you first shine the light on a male. What is the probability that you have entered room A? In other words, find .P(A|M) Seeing that we know the probability rule where P(M and A)=P(M∩A)=P(M)P(A│M), solving for P(A│M), we get

but here, we are stuck in a circular loop! We need to use Bayes’ Theorem to solve the problem:   represents the Bayes’ Theorem. So, Bayes’ Theorem can be generalized to:  

Hawai’ian Bar Card Trick

Other simple example applications are the Hawaiian bar card trick that I just made up or the old Monty Hall television game show. In the bar trick, suppose you take a nice trip to Maui and hang out at a local bar. Three hours later, you are bored and start thinking about a game to play with the attractive hostess. You take out three blank cards and write something down on each of the six sides. On the first card, you have Red and Red, the second card has Black and Black, and the third card has Red and Black. You then cover the cards with a napkin, and shuffle them, and proceed to ask the hostess to take one card out, showing only the top. Neither of you sees the bottom of the card. Now, suppose the top of the card says Red. You then insist that being the gentleman you are, you will also follow the young lady’s lead and select Red on the bottom. If the bottom is indeed Red, you win, but if it is Black, the hostess wins. You then proceed to tell a fake statistical tale. You say that since the top card is Red, there is no way it is the Black-Black card. Therefore, it must be either the Red-Red or Red-Black card, thereby having a 50:50 chance of being either Red or Black. And the payoff is that if you win, she brings you a free drink, but if she wins, you will double her tip (by the way, you have not yet told her what her tip was going to be, so, the double of nothing is still nothing, or the double of something fairly little is still little). The question is, what is the probability that you win or that she wins? In this situation, a Bayes-update calculation is needed, and we can boldly say that the chances of being Red or Black are not 50%. The calculations are shown next, however, before we jump into the math, it would be nice to understand the concept a little better and see if we can answer the question as to who wins (i.e., is the bottom of the card Red or Black) using basic logic. As mentioned, it can never be the Black-Black card. Hence, we are left with only the Red-Black or Red-Red card. Now, if you had selected the Red-Black card with the Red showing, then the other side must be Black. Score one for the Black team. Next, if she had selected the Red-Red card, you both could have been looking at the first side of the Red-Red card, or the second side of the Red-Red card. And in either scenario, the other side is Red, meaning there are two possible Red outcomes. Hence, the probability of Red at the bottom is 2/3 and the probability of Black at the bottom is 1/3.

Monty Hall Game Show

In the final segment of the old Monty Hall television game show, the host, Monty, would show the contestant three closed doors. Behind these doors were three rooms, where one room had a brand new car and the other two a Billy goat each. Clearly, to win the grand prize, the contestant must select the room with the shiny new car. Now, suppose the contestant selects Room 1. Monty then proceeds to the back and sees where the car is actually located. And he then opens a room door that was not previously chosen by the contestant, and which has a goat. Monty then gives the contestant a choice: stay with your current selection or switch to the last remaining room. The question is, should the contestant stay or switch? Which option provides a higher probability of winning? Let’s use some basic logic on this one. In the figure below, the car can be in any of the rooms and the other two rooms will be the goats. Now, suppose Room 1 was selected, and if the car is in Room 1, then staying will win and switching will lose the car. Score one for staying. In this case, Monty will open either Room 2 or Room 3 as both have Billy goats. However, if the car is in Room 2, Monte has no choice but to open Room 3 because Room 2 has the car and the contestant has already selected Room 1. Hence, switching will guarantee a win as the only room the contestant can switch to is Room 2. The same happens when the car is in Room 3, where Monty has no choice but to open Room 2, and, hence, if the switch is made, only Room 3 can be selected and it is a win. Therefore, there is a 2/3 probability of winning if the contestant switches and 1/3 if the contestant stays put.

COVID-19 TESTS: FALSE POSITIVES AND FALSE NEGATIVES

COVID-19 is highly contagious and at one point, 25% of the population has been infected. Suppose there is a low-priced rapid test kit that claims to have 99% accuracy in detecting the COVID-19 virus without PCR. Accuracy in this case means that if a person is truly sick, the test returns a positive result. However, the kit also has a possibility of false positives, creating mistakes in diagnosing healthy persons and identifying them as sick (i.e., the test shows a positive result even if the person is really not sick or infected). In the case of a person being tested with this kit shows a positive result, what is the probability that the person is really sick or infected, or shows negative when the person is, in fact, not sick or infected (correct diagnosis)? What is the probability of false positives and false negatives (incorrect diagnosis)? Is Type I or Type II error a bigger problem in this case? An interesting phenomenon occurs in this case. Rare diseases (low population with the sickness) even with higher levels of test accuracy tend to create a higher probability of false positives, but the false negatives are still relatively low. This is why it is so hard to diagnose exotic and uncommon illnesses. With a regular percentage of the population infected and sick (e.g., 25%), a lower accuracy test will yield significantly high false positives and false negatives. However, false positives usually outweigh false negatives in this situation.                  

The study of statistics refers to the collection, presentation, analysis, and utilization of numerical data to infer and make decisions in the face of uncertainty, where the actual population data is unknown. There are two branches in the study of statistics: descriptive statistics, where data is summarized and described, and inferential statistics, where the population is generalized through a small random sample, making it useful for making predictions or decisions when the population characteristics are unknown. A sample can be defined as a subset of the population being measured, while the population can be defined as all possible observations of interest of a variable. For instance, if one is interested in the voting practices of all U.S. registered voters, the entire pool of a hundred million registered voters is considered the population while a small survey of one thousand registered voters taken from several small towns across the nation is the sample. The calculated characteristics of the sample (e.g., mean, median, standard deviation) are termed statistics, while parameters imply that the entire population has been surveyed and the results tabulated. Thus, in decision making, the statistic is of vital importance considering that sometimes the entire population is yet unknown (e.g., who are all your customers, what is the total market share, and so forth) or it is very difficult to obtain all relevant information on the population because it would be too time- or resource-consuming. In inferential statistics, the following are the usual steps in conducting research:

• Designing the experiment––this phase includes designing the ways to collect all possible and relevant data.
  • Collection of sample data––data is gathered and tabulated
  • Analysis of data––statistical analysis performed
  • Estimation or prediction––inferences are made based on the statistics obtained
  • Hypothesis testing––decisions are tested against the data to see the outcomes
• Determining goodness-of-fit––actual data is compared to historical data to see how accurate, valid, and reliable the inference may be.
• Decision making––decisions are made based on the outcome of the inference.

MEASURING THE CENTER OF THE DISTRIBUTION––THE FIRST MOMENT

The first moments of the distribution of outcomes measure the expected rate of return on a particular project. They measure the location of the project’s scenarios and possible outcomes on average. The common statistics for the first moment include the mean (average), median (center of distribution), and mode (most commonly occurring value). Figure 3.1 illustrates the first moment––where in this case, the first moment of this distribution is measured by the mean (m) or average value. [caption id="attachment_7" align="aligncenter" width="351"] Figure 3.1: First Moment[/caption]

MEASURING THE SPREAD OF THE DISTRIBUTION––THE SECOND MOMENT

The second moment measures the spread of a distribution, which is a measure of risk. The spread or width of a distribution indicates the variability of a variable, that is, the potentiality that the variable can fall into different regions of the distribution––in other words, the potential scenarios of outcomes. Figure 3.2 illustrates two distributions with identical first moments (identical means) but very different second moments or risks. The visualization becomes clearer in Figure 3.3. As an example, suppose there are two stocks and the first stock’s movements (the solid line) with the smaller fluctuation is compared against the second stock’s movements (the dotted line) with a much higher price fluctuation. Clearly, an investor would view the stock with the wilder fluctuation as riskier because the outcomes of the riskier stock are relatively more unknown than the less risky stock. The vertical axis in Figure 3.3 measures the stock prices; thus, the riskier stock has a wider range of potential outcomes. This range is translated into a distribution’s width (the horizontal axis) in Figure 3.2, where the wider distribution represents the riskier asset. Hence, the width or spread of a distribution measures a variable’s risks. Notice that in Figure 3.2, both distributions have identical first moments or central tendencies but clearly, the distributions are very different. This difference in the distributional width is measurable. Mathematically and statistically, the width or risk of a variable can be measured through several different statistics, including the range, standard deviation (s), variance, coefficient of variation, percentile, interquartile range, confidence interval, volatility, beta, Value at Risk, and others. [caption id="attachment_8" align="aligncenter" width="288"] Figure 3.2: Second Moment[/caption]   [caption id="attachment_9" align="aligncenter" width="367"] Figure 3.3: Stock Price Fluctuations[/caption]

Variance and Standard Deviation

Variance and standard deviation are two common measures of the second moment. Variance is the average of the squared deviations about their means, in squared units:   Standard deviation is in original units and, thus, useful as a direct means of comparison of dispersion and variability measured in the same units: Although standard deviation and variances have many uses, those uses are limited because their measurements are in the same units and, hence, are considered absolute values of risk, uncertainty, or spread. Greek letters (μ, σ) and upper-case letters (N) represent the population whereas standard Latin alphabets and lower-case letters (s, n, x) represent the sample.

Coefficient of Variation

The coefficient of variation (CV) is unitless and measures relative variability. It thus allows the comparison of two datasets to see which has more variability without worrying about the units.

CV = s/ x̅ or CV = /

Statistic	# in family	Food expenditure ($)
x̅	3.23	$110.5
s	1.34	$25.25

  Which has more variation, the number of family members or the food expenditure?  CV in family = 1.34/3.23 = 0.415 CV in expenditures = 25.25/110.25 = 0.229   The calculations show that there is more variation in the number of family members.

MEASURING THE SKEW OF THE DISTRIBUTION––THE THIRD MOMENT

The third moment measures a distribution’s skewness, that is, how the distribution is pulled to one side or the other. Figure 3.4 illustrates a negative or left skew (the tail of the distribution points to the left) and Figure 3.5 illustrates a positive or right skew (the tail of the distribution points to the right). The mean is always skewed towards the tail of the distribution, while the median remains constant. Another way of seeing this is that the mean moves but the standard deviation, variance, or width may still remain constant. If the third moment is not considered, then looking only at the expected returns (mean) and risk (standard deviation), a positively skewed project might be incorrectly chosen! For example, if the horizontal axis represents the net revenues of a project, then clearly, a left or negatively skewed distribution might be preferred as there is a higher probability of greater returns (Figure 3.4) as compared to a higher probability for lower-level returns (Figure 3.5). Thus, in a skewed distribution, the median is a better measure of returns, as the medians for both Figures 3.4 and 3.5 are identical, risks are identical, and, hence, a project with a negatively skewed distribution of net profits is a better choice. Failure to account for a project’s distributional skewness may mean that the incorrect project may be chosen (e.g., two projects may have identical first and second moments, that is, they both have identical returns and risk profiles, but their distributional skews may be very different).   [caption id="attachment_15" align="aligncenter" width="577"] Figure 3.4: Third Moment (Left Skew)[/caption]   [caption id="attachment_16" align="aligncenter" width="577"] Figure 3.5: Third Moment (Right Skew[/caption]  

MEASURING THE CATASTROPHIC TAIL EVENTS IN A DISTRIBUTION––THE FOURTH MOMENT

The fourth moment, or kurtosis, measures the peakedness of a distribution. Figure 3.6 illustrates this effect. The background is a normal distribution with a kurtosis of 3.0. The new distribution has a higher kurtosis, thus the area under the curve is thicker at the tails with less area in the central body. This condition has major impacts on risk analysis because the two distributions in Figure 3.6, the first three moments (mean, standard deviation, and skewness) can be identical but the fourth moment (kurtosis) is different. This means that although the returns and risks are identical, the probabilities of extreme and catastrophic events (potential large losses or large gains) occurring are higher for a high kurtosis distribution (e.g., stock market returns are leptokurtic or have high kurtosis). Ignoring a project’s return’s kurtosis may be detrimental. Fortunately, the calculations for these four moments are automatically done for you using the Risk Simulator software as will be seen in later chapters.   [caption id="attachment_20" align="aligncenter" width="530"] Figure 3.6: Fourth Moment[/caption]  

Most distributions can be defined up to four moments. The first moment describes its location or central tendency (expected returns); the second moment describes its width or spread (risks); the third moment, its directional skew (most probable events); and the fourth moment, its peakedness or thickness in the tails (catastrophic losses or gains). All four moments should be calculated and interpreted to provide a more comprehensive view of the project under analysis. Finally, the term moment refers to the highest power of x in each of the statistics’ equations.

THE BASICS OF INTERPRETING PDF, CDF, & ICDF CHARTS

This appendix briefly explains the probability density function (PDF) for continuous distributions, which is also called the probability mass function (PMF) for discrete distributions (we use these terms interchangeably), where given some distribution and its parameters, we can determine the probability of occurrence given some outcome or random variable x. In addition, the cumulative distribution function (CDF) can also be computed, which is the sum of the PDF values up to this x value. Finally, the inverse cumulative distribution function (ICDF) is used to compute the value x given the cumulative probability of occurrence.  In mathematics and Monte Carlo risk simulation, a probability density function (PDF) represents a continuous probability distribution in terms of integrals. If a probability distribution has a density of f(x), then, intuitively, the infinitesimal interval of [x, x + dx] has a probability of f(x)dx. The PDF, therefore, can be seen as a smoothed version of a probability histogram; that is, by providing an empirically large sample of a continuous random variable repeatedly, the histogram using very narrow ranges will resemble the random variable’s PDF. The probability of the interval between [a, b] is given by , which means that the total integral of the function f must be 1.0.  It is a common mistake to incorrectly think of f(a) as the probability of a. In fact, f(a) can sometimes be larger than 1 (consider a uniform distribution between 0.0 and 0.5). The random variable x within this distribution will have f(x) greater than 1. The probability, in reality, is the function f(x)dx discussed previously, where dx is an infinitesimal amount. 

The cumulative distribution function (CDF) is denoted as F(x) = P(X ≤ x), indicating the probability of X taking on a less than or equal value to x. Every CDF is monotonically increasing, is continuous from the right, and at the limits has the following properties: and

Further, the CDF is related to the PDF by, where the PDF function f is the derivative of the CDF function F. In probability theory, a probability mass function, or PMF, gives the probability that a discrete random variable is exactly equal to some value. The PMF differs from the PDF in that the values of the latter, defined only for continuous random variables, are not probabilities; rather, its integral over a set of possible values of the random variable is a probability. A random variable is discrete if its probability distribution is discrete and can be characterized by a PMF.  Therefore, X is a discrete random variable if  as u runs through all possible values of the random variable X.

INTERPRETING PROBABILITY CHARTS

Here are some tips to help decipher the characteristics of a distribution when looking at different PDF and CDF charts:

• For each distribution, a continuous distribution’s PDF is shown as an area chart (Figure A.1) whereas a discrete distribution’s PMF is shown as a bar chart (Figure A.2). 
• If the distribution can only take a single shape (e.g., normal distributions are always bell shaped, with the only difference being the central tendency measured by the mean and the spread measured by the standard deviation), then typically only one PDF area chart will be shown with an overlay PDF line chart (Figure A.3) showing the effects of various parameters on the distribution. 
• The CDF charts, or S-Curves, are shown as line charts (Figure A.4), and sometimes as bar graphs. 
• The central tendency of a distribution (e.g., the mean of a normal distribution) is its central location (Figure A.3).
• Multiple area charts and line charts will be shown (e.g., beta distribution) if the distribution can take on multiple shapes (e.g., the beta distribution is a uniform distribution when alpha = beta = 1; a parabolic distribution when alpha = beta = 2; a triangular distribution when alpha = 1 and beta = 2, or vice versa; a positively skewed distribution when alpha = 2 and beta = 5, and so forth). In this case, you will see multiple area charts and line charts (Figure A.5). 
• The starting point of the distribution is sometimes its minimum parameter (e.g., parabolic, triangular, uniform, arcsine, etc.) or its location parameter (e.g., the beta distribution’s starting location is 0, but a beta 4 distribution’s starting point is the location parameter; Figure A.5 shows a beta 4 distribution with location = 10, its starting point on the x-axis).
• The ending point of the distribution is sometimes its maximum parameter (e.g., parabolic, triangular, uniform, arcsine, etc.) or its natural maximum multiplied by the factor parameter shifted by a location parameter (e.g., the original beta distribution has a minimum of 0 and maximum value of 1, but a beta 4 distribution with location = 10 and factor = 2 indicates that the shifted starting point is 10 and ending point is 11, and its width of 1 is multiplied by a factor of 2, which means that the beta 4 distribution now will have an ending value of 12, as shown in Figure A.5).
• Interactions between parameters are sometimes evident. For example, in the beta 4 distribution, if the alpha = beta, the distribution is symmetrical, whereas it is more positively skewed the greater the difference between beta and alpha, and the more negatively skewed, the greater the difference between alpha and beta (Figure A.6).
• Sometimes a distribution’s PDF is shaped by two or three parameters called shape, scale, and location. For instance, the Laplace distribution has two input parameters, alpha location and beta scale, where alpha indicates the central tendency of the distribution (like the mean in a normal distribution) and beta indicates the spread from the mean (like the standard deviation in a normal distribution).
• The narrower the PDF (Figure A.3’s normal distribution with a mean of 10 and standard deviation of 2), the steeper the CDF S-Curve looks (Figure A.4), and the smaller the width on the CDF curve.
• A 45-degree straight line CDF (an imaginary straight line connecting the starting and ending points of the CDF) indicates a uniform distribution; an S-Curve CDF with equal amounts above and below the 45-degree straight line indicates a symmetrical and somewhat bell- or mound-shaped curve; a CDF completely curved above the 45-degree line indicates a positively skewed distribution (Figure A.7), while a CDF completely curved below the 45-degree line indicates a negatively skewed distribution (Figure A.8).
• A CDF line that looks identical in shape but shifted to the right or left indicates the same distribution but shifted by some location, and a CDF line that starts from the same point but is pulled both to the left and right indicates a multiplicative effect on the distribution such as a factor multiplication, as shown in Figures A.9 and A.10.
• An almost vertical CDF indicates a high kurtosis distribution with fat tails, and where the center of the distribution is pulled up (e.g., see the Cauchy distribution) versus a relatively flat CDF, a very wide and perhaps flat-tailed distribution is indicated. 
• Some discrete distributions can be approximated by a continuous distribution if its number of trials is sufficiently large and its probability of success and failure is fairly symmetrical (e.g., see the binomial and negative binomial distributions). For instance, with a small number of trials and a low probability of success, the binomial distribution is positively skewed, whereas it approaches a symmetrical normal distribution when the number of trials is high, and the probability of success is around 0.50. 
• Many distributions are both flexible and interchangeable–refer to the details of each distribution––e.g., binomial is Bernoulli repeated multiple times; arcsine and parabolic are special cases of beta; Pascal is a shifted negative binomial; binomial and Poisson approach normal at the limit; chi-square is the squared sum of multiple normal; Erlang is a special case of gamma; exponential is the inverse of the Poisson but on a continuous basis; F is the ratio of two chi-squares; gamma is related to the lognormal, exponential, Pascal, Erlang, Poisson, and chi-square distributions; Laplace comprises two exponential distributions in one; the log of a lognormal approaches normal; the sum of multiple discrete uniforms approach normal; Pearson V is the inverse of gamma; Pearson VI is the ratio of two gammas; PERT is a modified beta; a large degree of freedom T approaches normal; Rayleigh is a modified Weibull; and so forth.

[caption id="attachment_23" align="aligncenter" width="638"] Figure A.1: Continuous PDF (Area Chart)[/caption]   [caption id="attachment_24" align="aligncenter" width="638"] Figure A.2: Discrete PMF (Bar Chart)[/caption]   [caption id="attachment_25" align="aligncenter" width="638"] Figure A.3: Multiple Continuous PDF Overlay Charts[/caption]   [caption id="attachment_26" align="aligncenter" width="638"] Figure A.4: CDF Overlay Charts[/caption]   [caption id="attachment_27" align="aligncenter" width="638"] Figure A.5: PDF Characteristics of the Beta Distribution[/caption]   [caption id="attachment_28" align="aligncenter" width="638"] Figure A.6: PDF of a Negatively Skewed Beta Distribution[/caption] [caption id="attachment_29" align="aligncenter" width="638"] Figure A.7: CDF of a Positively Skewed Distribution[/caption]   [caption id="attachment_30" align="aligncenter" width="638"] Figure A.8: CDF of a Negatively Skewed Distribution[/caption] [caption id="attachment_31" align="aligncenter" width="638"] Figure A.9: PDF Characteristics of a Shift[/caption]   [caption id="attachment_32" align="aligncenter" width="638"] Figure A.10: CDF Characteristics of a Shift[/caption]