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The result that fXY (x, y; X, Y, X, Y, ) integrates to 1 is left as an exercise. Also, the bivariate normal probability density function is positive over the entire plane of real numbers. Two examples of bivariate normal distributions are illustrated in Fig. 5-17 along with corresponding contour plots. Each curve on the contour plots is a set of points for which the probability density function is constant. As seen in the contour plots, the bivariate normal probability density function is constant on ellipses in the (x, y) plane. (We can consider a circle to be a special case of an ellipse.) The center of each ellipse is at the point ( X, Y). If 0 ( 0), the major axis of each ellipse has positive (negative) slope, respectively. If 0, the major axis of the ellipse is aligned with either the x or y coordinate axis. EXAMPLE 5-33 1 2 2 e 0.51x y 2 is a special case of a bivariate 12 normal distribution with X 1, Y 1, X 0, Y 0, and 0. This probability density function is illustrated in Fig. 5-18. Notice that the contour plot consists of concentric circles about the origin. The joint probability density function fXY 1x, y2 By completing the square in the exponent, the following results can be shown. The details are left as an exercise.

Camara et al. s (2000) collected survey data on test usage and assessment from 933 clinical psychologists and 567 neuropsychologists who were randomly selected from the American Psychological Association (APA) and the National Association of Neuropsychology (NAN). The authors were interested in data from practitioners who conducted assessments on a regular basis, so they ultimately conducted their analyses on data from respondents who engaged in 5 or more hours per week of assessment-related services. Thus, the final sample used for ranking test usage comprised 179 clinical psychologists (19% of the clinical psychologist respondents) and 447 neuropsychologists (79% of the neuropsychologist respondents). Table 1.1 displays the hours spent administering, scoring, and interpreting psychological tests during a typical week, for the total number of respondents to the survey (N = 1,500).

y f XY (x, y) f XY (x, y)

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It facilitates the conscious selection of AT&L requirements, so that decisions about selecting AT&L mechanisms have a clear basis, rather than occurring in a vacuum. It promotes explicit analysis of trade-offs that encourages balancing and prioritizing of conflicting requirements and forces. This includes balancing the need for accountability with the need for privacy. This helps to avoid stronger than necessary AT&L mechanisms that would make it difficult for valid users, and at the same time it helps to avoid weaker than necessary AT&L that makes it easy for unauthorized actors to avoid. It results in documentation of AT&L requirements that communicates to all interested parties and is useful in comparing the adequacy of alternative implementations of AT&L services. The explicit requirements resulting from the pattern foster a clear connection of requirements to audit and logging policies: this also encourages organizations to make their accounting policies more explicit.

Hours spent administering, scoring, and interpreting psychological tests during Clinical Psychologists Neuropsychologists n (%) 116 (20.5) 62 (10.9) 92 (16.2) 105 (18.5) 188 (33.2) 4 (<1) 567 (100.0) Cumulative % 100.0 78.8 67.9 51.7 33.2 <1

Figure 5-17

f XY (x, y)

n (%) 755 (80.9) 62 (6.6) 39 (4.2) 36 (3.9) 37 (4.0) 4 (<1) 933 (100.0)

The following potential liabilities may arise from applying this pattern:

If X and Y have a bivariate normal distribution with joint probability density fXY (x, y; X, Y, X, Y, ), the marginal probability distributions of X and Y are normal with means X and Y and standard deviations X and Y, respectively. (5-33)

This chapter examines individual differences in intelligence on the important background variables of gender, ethnicity, urban rural residence, occupational category, and educational attainment; it then relates these differences to comparable findings in the literature. Recent data on the WAIS-III are featured, along with recent data obtained on the KAIT and other Kaufman tests, and the current data are integrated with data obtained in earlier generations on the WAIS-R and WAIS, and, when pertinent, with Stanford-Binet IV data for adolescents and

Figure 5-19 illustrates that the marginal probability distributions of X and Y are normal. Furthermore, as the notation suggests, represents the correlation between X and Y. The following result is left as an exercise.

It requires an investment of resources to apply the pattern, including time to analyze domains and AT&L needs. In some cases the cost of applying the pattern may exceed its benefits. It poses a danger of possible violation of privacy rights if extensive data is captured and analyzed. You can mitigate this by capturing and analyzing the minimum amount of data, and by working closely with your legal department. The formal selection process may be too long and costly and produce too much overhead. You can mitigate this in the same ways as noted above. Specific circumstances might not be covered by generic AT&L requirements. You can mitigate this by adding specific requirements and including them in the trade-offs. Documentation of requirements implies that they must be maintained as they change over time. You can mitigate this by keeping the requirements in a form that is easy to update, integrated with other system documentation.

If X and Y have a bivariate normal distribution with joint probability density function fXY (x, y; X, Y, X, Y, ), the correlation between X and Y is . (5-34)

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