A CLASS OF MESOKURTIC DISTRIBUTIONS AS ALTERNATIVE TO NORMAL DISTRIBUTION

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1 CHAPTER 3 A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL 3. Introduction This chapter deals with a comparative study between normal distribution and the new class of distributions given by (2.5.2) in chapter 2. We also examine the myths and realities about Pearson s moment measure of kurtosis β 2. In section 3.2, we examine β 2 as a measure of tail length, tail weight and lack of shoulders and shows that it does 33

2 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL not measure any of them. In section 3.3, we consider Horn s (983) measure of kurtosis and show that it properly in many situations detects shifts from normality even when Peasrson s moment measure of kurtosis β 2 is unchanged. In section 3.4, we consider the concept of spread function and examine its use in kurtosis ordering. In section 3.5, we consider Van Zwet s (964) measure to order symmetric distributions and its use in ordering the new class of symmetric mesokurtic distributions. Section 3.6, considers a few other kurtosis ordering techniques and examine their uses in kurtosis ordering. Here we introduce an ordering based on moments and use it to order the family of distributions given by (2.5.2). Section 3.7, considers the efficiency of sample mean as estimator of population mean when the sample is taken from the class of distributions given by (2.5.2). A simulation study is conducted using Monte Carlo runs and the results of which are presented in table A MATLAB program for the simulation study is presented in the appendix to this chapter. 3.2 Kurtosis as a Measure of Peakedness, Tail Weight, Tail Length and Lack of Shoulders We have seen that Pearson introduced β 2 as a measure of peakedness at the center of a distribution and closeness of β 2 to three as an indication of near normality. Many tests were developed to test normality based on the value of β 2. Among the Pearsonian system of frequency curves normal is the only distribution with kurtosis β 2 = 3. But Hildebrand s example made a change about the concept of β 2. Ever since Hildebrand s example, the interpretation of Pearson s moment measure of kurtosis β 2 was a matter of debate. There seems to be no universal agreement about the meaning and interpretation of kurtosis. Darlington (97) argued that β 2 is a measure of bimodality of a distribution and showed that if Z is a standardized random variable, then β 2 = V ar(z 2 ) +. (3.2.) 34

3 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL Hence β 2 can be viewed as a measure of the dispersion of Z 2 around its expectation, or equivalently, the dispersion of Z around the values - and +. Moors (986) showed that even though the derivation (3.2.) is correct, the inference about the measure of bimodality is not true, as bimodal distributions have large kurtosis, only if modes are not close to the points Z = ±. He argued that the kurtosis measures the dispersion around the two values µ ± σ of a random variable X having mean µ and variance σ 2. While few think that kurtosis measures the peakedness at the center of a distribution (at least in the unimodal case), few others think that it measures heaviness of the tails or the length of the tails. Motivated by Hildebrand s example and Kale and Sebastian (996) in this chapter we make a closer look at Pearson s moment measure of kurtosis and its interpretations. There is no agreement among statisticians on what the kurtosis coefficient really measures. Pearson (95) introduced kurtosis as a measure of peakedness at the center of a symmetric distribution compared to a normal density. He explained that if a symmetric unimodal distribution f(x) is more peaked than a normal distribution (assuming same variance and center of symmetry) necessarily f(x) should have β 2 > 3 and he termed the distribution as leptokurtic. Similarly he termed that for a symmetric unimodal distribution which is more flat-topped than normal as platykurtic. While Laplace and logistic distributions are examples of leptokurtic distributions, uniform is an example of a platykurtic distribution. We show here that even when the distributions are unimodal β 2 does not measure the height at the center of a distribution. Consider the pdf given by g(y) = 6 6 exp( 2/2 y ) I (, ) (y) ( 7 4 y 2 )I( 7, 7) (y) (3.2.2) 7 and f(y) = [ + 8 H 6(y)]φ(y) < y <. (3.2.3) Here I in the equation (3.2.2) denotes the indicator function. We note that both the den- 35

4 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL sities given above are symmetric about zero with unit variance and Pearson s measure of kurtosis β 2 = 3. Figure (3.2.) given below shows the plot of g(y), f(y) and a standard normal. It shows that even though the kurtosis measure is the same for three densities, peakedness of g(y) at the center is higher than standard normal but peakedness of f(y) at the center is less than that of a standard normal indicating that β 2 is having a very little impact on the peakedness of the pdf..7 Fig(3.2.) Plot of three symmetric mesokurtic densities having same variance.6 g(x).5.4 N(,) f(x) Many authors including Ruppert (987) argue that kurtosis measures tail weight of a density as, if one moves probability mass from flanks to the center a distribution, to keep scale fixed one must also move mass from flanks to the tail. That is, if a distribution is less peaked at the center than normal, it should be a short tailed distribution than normal and if a distribution is more peaked at the center than normal, it should be long tailed one than normal. The following table gives P [X > 3] for different values of α from the class of densities given in (2.5.2) along with standard normal (α = ). When the value of α increases the peakedness of the pdf at the center decreases in this family. But the values in the table 3.2. show that even when the peakedness decreases the P [X > 3] increases. It shows that we cannot associate tail weight or length with β 2. 36

5 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL Table 3.2. P (X > 3) α=.3 α=.5 α=2.6 α=3.7 α=4.8 α=5.9 α=6.2 α=6.5.2 The pdf s in (2.5.2) posses some interesting properties and they can be used as possible alternatives to normal model in robustness studies and regression analysis. We have several situations in which a wider class such as the symmetric mesokurtic family of distributions, can be used as an alternative to Gaussian model. For example, such a class of distributions can be used to model data when so called outliers and inliers are present in it by considering it as a valuable information about non normality. Another possible application of the model is, when the in log returns of share prices, stock indices and foreign exchange rates, the frequency of large and small changes, relative to the range of data, is rather high which leads us to believe that the data do not come from normal. This is due to the fact that, large and small values in a log return sample tend to occur in clusters. Since any test on normality based on β and β 2 will have zero power in this class it will be interesting to see how probability plots will behave to random samples from this class. We have selected a random sample of size 5 from the model (2.5.2) for α =, α = 4 and α = 6 and the normal QQ plots are constructed. 37

6 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL Fig (3.2.2) 4 QQ Plot of Sample Data of f(x) for α= versus Standard Normal 4 QQ Plot of Sample Data of f(x) for α=4 versus Standard Normal Quantiles of Input Sample Quantiles of Input Sample Standard Normal Quantiles Standard Normal Quantiles QQ Plot of Sample Data of f(x) for α=6 versus Standard Normal 3 2 Quantiles of Input Sample Standard Normal Quantiles The graph shows that from a normal QQ plots it is difficult to detect shifts from normality or multimodality in the class of densities given by (2.5.2). One can also note that very few points doesn t fall in the linearity pattern like outliers or inliers and there by indicating that the samples from the data behave like normal model with outliers or inliers. 3.3 Horn s Measure of Kurtosis The peakedness measure introduced by Horn (983) is applicable to a symmetric density with pdf f(.) and corresponding distribution F(.). Let us define, the inverse F (p) by F (p) = inf{x : F (x) p} for < p <. Now we consider a rectangle with lines 38

7 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL x =, y =, y = f() and x = F (p +.5) for some < p <.5. Then the area of the rectangle is A p (f) = f()f (p +.5). The area under the density in the above mentioned rectangle is then equal to p for all f(.). Horn took m p (f) = p/a p (f) as a measure of kurtosis as, p/a p (f) can be considered as the proportion of area covered by f(.) in the rectangle. If p/a p (f) is nearer to (m p (f) = ), then most of the density is under the rectangle and therefore f(.) is looking like a rectangle and hence not very peaked. If p/a p (f) is nearer to, then f(.) looks like a spike with a long tail and hence peaked. The following figure shows the graph of m p (f) for different values of α in the densities given by (2.5.2). The Fig (3.3.) indicates that Horn s measure is sensitive to peakedness even when there is no difference in β 2 values. It is able to differentiate distributions even when Pearson s measure fails to do so. It also has the advantage that it exists even when moments of the distribution do not exist. Curves in the graph show that in terms of peakedness, the densities are in the descending order as α increases, where α < 6.9. Fig (3.3.) Horn s measure of kurtosis m(f) Normal alpha= alpha=2 alpha=3 alpha=4 alpha=5 alpha= p 3.4 Spread Function Another important concept in measuring peakedness, tail weight and kurtosis is, the spread function. For example we refer to Bickel and Lehmann (979). For a distribution 39

8 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL F(.), the spread function S F (.) is defined by S F (u) = F (.5 + u) F (.5 u) for u < 2. (3.4.) Since equation (2.5.2) is symmetric, the spread function for the class of densities in (2.5.2) will reduce to the following form S FX (u) = 2F X (.5 + u) for u < 2. (3.4.2) Note that the spread function is location invariant and for any real a and for a random variable X, S FaX (u) = a S FX (u), for u < 2, where, S F ax (u) and S FX (u) are spread functions of ax and X respectively. The spread function of a distribution describes the manner in which probability mass is placed symmetrically about its median and hence can be used to formulate concepts such as peakedness, tail weight etc. For example S FX (u) is large for u near ; then, compared to inter-quartile range, the tails of F are S FX ( 4 ) 2 stretched. See Balanda and MacGillivray (988). 3.5 Van Zwet s Ordering Van Zwet (964) introduces the concept of ordering of two distributions F and G which is denoted by the symbol F s G and holds if and only if the spread function S G (S F (s)) is convex for s. For two symmetric distributions F and G, this definition reduces to, F s G if and only if G (F (x)) is convex for x > m F and concave for x < m F, where m F is the median of F. Fig (3.5.) given below shows the plots of Φ (F α (x)) against x, for x > and for different values of α. The graphs show that Van Zwet s ordering is not possible for the distributions in the class of symmetric mesokurtic distributions given by (2.5.2) with respect to the standard normal distribution. 4

9 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL Fig(3.5.) 3 alpha= 3 alpha=2 3 alpha= alpha= alpha= alpha= Since we cannot establish any convexity from the graph, we conclude that the distributions F and G, where F follows (2.5.2), the transformation from F to Φ doesn t show any contraction of the inter-quartile distance S F (u) around the median or an extension of the distances when we move away from zero (median). 3.6 Other Kurtosis Comparisons Other weaker orderings can also be used in kurtosis comparisons. For example star orderings by Bruckner and Ostrow (962), F γ G if and only if S G(u) S G (γ) S H (u) S H is increas- (γ) ing for u <. Even though the strongest ordering is not possible for the above 2 distributions, the graph indicates that some partial orderings may be possible for these distributions. For a detailed discussion of other weaker ordering we refer to McGillivray and Balanda (99). There are some other quantile based measures, which appeared in the literature. Doksum (969), Lawrence (975), Riverst (982), Benjami (983) and 4

10 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL Loh (982) have made such attempts. For example Lawrence defined an ordering R for two symmetric distributions by F R G if and only if G (p) m G F (p) m F is increasing for < p < 2 (decreasing for 2 < p < ). We have plotted the graph of G (p) F (p) (3.6.) for 2 α=,2,5 and 6. in Fig < p <, where F is the distribution function corresponding to (2.5.2), for Fig(3.6.).4 Plot of Lawrence function for alpha= against Normal.6 Plot of Lawrence function for alpha=2 against Normal Plot of Lawrence function for alpha=5 against Normal Plot of Lawrence function for alpha=6 against Normal The graphs given in Fig (3.6.) show that F R Φ is not possible in general for this family, but if we take a partial ordering in the tail area such orderings can be possible. Another tail weight measure considered in the literature for any distribution F is t γ,δ (F ) = F (.5 + γ) F (.5 γ) F (.5 + δ) F (.5 δ) (3.6.) 42

11 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL for < δ < γ < 2. For a symmetric distribution, t γ,δ(f ) becomes t γ,δ (F ) = F (.5 + γ) F (.5 + δ). (3.6.2) We have calculated the tail measure for the class of densities in (2.5.2) for different values of α are given in Table 3.6. along with standard normal. Table 3.6. Normal α = α = 2 α = 3 α = 4 α = 5 α = 6 t 3 8, 4 t 7 6, 4 t 7 6, One can observe from the table that the tail weight measure t 7 6, 4 between the normal desity and the class of densities given by (2.5.2). well distinguishes 3.6. Stretched and Squeezed Distributions. The ordering given by Lawrence (975) can be viewed in another way using stretched and squeezed distributions. Definition 3.6. Let F and G be two symmetric distributions. F is stretched with respect to G (F > st G), if F (p) F ( 2 ) G (p) G ( 2 ) is increasing in p for 2 < p <. F is squeezed with respect to G, if F (p) F ( 2 ) G (p) G ( 2 ) is decreasing in p for 2 < p <. For any two distributions F and G, symmetric about zero, the above definition implies that F > st G, if F (p) is increasing. Equivalently we have G (p) F (p) = F (p) G (p) G (p). (3.6.3) Thus F can be obtained from G through multiplication by an increasing function, or the shape of G is stretched to get the shape of F. In other words, if F > st G, then F is scaled to match G at some x o > ( F (x o ) = G(x o )), and then tails of F are heavier than tails of G. That is 43

12 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL F (x) G(x) x > x o. (3.6.4) Even though the distributions in the class (2.5.2) cannot be considered as stretched or squeezed with respect to normal, it is interesting to note that we can find an x o independent of α, such that the condition (3.6.4) is satisfied. If F is in (2.5.8) and G = Φ, the standard normal cdf, the condition (3.6.4) implies that Φ(x o ) α 72 H 5(x o )φ(x o ) Φ(x o ) for < x o <. (3.6.5) Routine calculations show that, we can find a value x o = 5 + (3.6.6) such that the condition (3.6.4) is satisfied An Ordering Using Moments Theorem 3.6. If F is the distribution function given in (2.5.8) and Φ is the standard normal distribution function then, all even order moments of F are greater than or equal to the corresponding moments of Φ. Proof Since F and Φ are symmetric about zero all odd order moments are equal to zero. Further, since they have the same β 2, their first four moments are equal. Now, even order moments of standard normal are given by µ 2r Φ =.3...(2r ), r =, 2,... (3.6.7) 44

13 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL Even order moments of F are given by µ 2r F =.3...(2r ) [ + k((2r + )(2r + 3)(2r + 5) 5(2r + )(2r + 3) + 45(2r + ) 5)], (3.6.8) where, k = α and hence, 72 µ 2r F µ 2r Φ = 8kr(r )(r 2), r = 3, 4,... (3.6.9) Since k is positive, equation (3.6.9) is positive for all r > 2. That is, µ 2r F µ 2r Φ > r > 2. That is all even order moments greater than four of F will be greater than that of Φ. Equation (3.6.9) shows that the difference in moments of this class with standard normal is a functions of k. Hence, we can order the distributions in the class (2.5.2) using a moment ordering. Equation (3.6.9) implies that all moments of the pdfs in this class are functions of α ( α 6.9), which is zero for the normal. Therefore one possible ordering of the distributions in this class is Φ < m F. Definition Two distributions F and G, symmetric about µ is said to be G m F, if all moments of F are greater than or equal to the corresponding moments of G. 45

14 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL 3.7 Efficiency of Mean as an Estimator of Location Parameter Sebastian and James (22) showed that the pdf g(y) given by (3.2.2) is an example of a symmetric mesokurtic distribution, for which asymptotic efficiency of median is greater than mean as location estimator. But the class of distributions given in (2.5.2) gives examples of distributions in symmetric mesokurtic class with asymptotic efficiency of mean as an estimator of µ, is greater than that of median. This result is shown below: Let M n be the median of the sample (X, X 2,..., X n ), then V ar(m n ) = 4nf 2 µ where, f µ is the median ordinate. From (2.6.), we have, f µ = 2πσ [ 5α 72 ]. Then V ar(m n ) = π σ2 2( 5α )2 n. 72 It is easy to verify that V ar(m n ) = π σ 2 2( 5α 72 )2 n > σ2 n for α < 6.9. Therefore asymptotic efficiency of mean will be greater than that of median as an estimator of the location parameter. This is also evident from the simulation results given in the table Godambe and Thompson (989) showed that in a semi-parametric set-up in which first four moments are known functions of µ and σ 2, the optimal estimating equations for (µ, σ 2 ) based on a sample (X, X 2,..., X n ) coincide with that of the likelihood equations for N(µ, σ 2 ) whenever β = and β 2 = 3 and lead to ˆµ = X and ˆσ 2 = n (Xi X) 2. In Table 3.7. we give the result of simulation based on random samples taken from a family of pdf s given by 46

15 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL f(x, α, µ, σ) = σ [ + α 72 H 6 ( x µ σ )] φ ( x µ σ ), < x <. Random samples are generated from pdf for α =, 2,..., 6 and ˆµ and ˆσ 2 are calculated for n=, 2, 5 and. Simulation is done for N= times and average values of ˆµ and ˆσ 2 are given in the table along with their standard error reported in brackets. Table 3.7. α = α = α = 2 α = 3 α = 4 α = 5 α = 6 ˆµ (.) (.) (.959) (.3) (.8) (.9) (.995) n= Median (.397) (.439) (.435) (.495) (.589) (.667) (.638) ˆ σ (.826) (.824) (.745) (.77) (.847) (.759) (.77) ˆµ (.497) (.489) (.489) (.496) (.497) (.497) (.497) n=2 Median (.752) (.75) (.778) (.822) (.853) (.855) (.89) ˆ σ (.92) (.946) (.945) (.84) (.938) (.94) (.948) ˆµ (.94) (.97) (.2) (.98) (.97) (.97) (.2) n=5 Median (.36) (.329) (.339) (.347) (.37) (.387) (.399) ˆ σ (.378) (.379) (.398) (.39) (.377) (.376) (.393) ˆµ (.) (.) (.) (.) (.) (.) (.) n= Median (.6) (.67) (.76) (.84) (.89) (.9) (.98) ˆ σ (.23) (.22) (.23) (.23) (.22) (.22) (.22) 3.8 Conclusion The concept of kurtosis seems to be rather difficult to interpret. We have shown the inability of the kurtosis measure β 2 in measuring what it was supposed to do. While 47

16 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL the example introduced by Hildebrand (97) showed the inability of β 2 as a measure of bimodality, the example introduced by Kale and Sebastian (996) and Sebastian and James (22) showed the inability of β 2 as a measure of peakedness, tail weight, lack of shoulders, or even multimodality. Except few kurtosis measures like Horn s measure, most of the available kurtosis measures were unable to fully detect the shifts in peakedness or tail weight in the class of densities given by (2.5.2). Our Simulation experiment shows that even in small samples, sample mean has lesser mean squared error compared to that of the sample median as an estimator of the location parameter θ. APPENDIX 3. A Matlab program for the computation of standard errors of sample median, sample mean and sample variance from KS class, based on the simulated sample. r = 5; n = ; alpha = 6; k = alpha/72; sum = ; ssvar = ; sssum = ; me = zeros(r, ); forj = : r; ss = ; s = ; ssq = ; summe = ; sssumme = ; fori = : n; u = rand(, ); a = 5; b = 5; c = (a + b)/2; ind = ; while ind > fnx = cdf( Normal, c,, ) k (c 5 c c) pdf( Normal, c,, ); if abs(fnx u) <.; ind = ; else iffnx < u; a = c; 48

17 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL c = (a + b)/2; else b = c; c = (a + b)/2; end; end; end; y(i) = c; s = s + y(i); ss = ss + y(i) y(i); end; p = n/2; q = p + ; z = sort(y); me(j) = (z(p) + z(q))/2; m(j) = s/n; var(j) = (/n) ss m(j) m(j); j end; forj = : r; sum = sum + m(j); summe = summe + me(j); sssum = sssum + m(j) m(j); sssumme = sssumme + me(j) me(j); ssq = ssq + var(j); ssvar = ssvar + var(j) var(j); end; 49

18 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL mean = sum/r varmean = (sssum/r) mean 2 meanme = summe/r varme = (sssumme/r) meanme 2 variance = ssq/r varvar = (ssvar/r) variance 2 5

19 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL References. Balanda, K.P. and MacGillivray, H.L. (988). Kurtosis: A critical review. The Amer. Statis Benjami, Y. (983). Is the t test really conservative when the parent distribution is long tailed. J. Amer. Statist. Assoc Bickel. P.J. and Lehmann, E.L.(979). Descriptive Statistics for Non-parametric Models. IV. Spread. Contributions to Statistics. Jaroslav Hajek Memorial Volume (J. Jureckova. ed.) Academia, Prague Bruckner, A.M. and Ostrow, E. (962). Some function classes related to the class of convex functions. Pasefic. J. Math D Agostino, R. B., and Tietjen, G.L. (973). Approaches to the null distribution of b. Biometrika Darlinton, R.B. (97). Is kurtosis really peakedness. The Amer. Statist Doksum. K.A. (975). Measures of location and symmetry. Scand. J. Statist Godambe, V.P., Thompson, M.E. (989). An extension of quasi-likelihood estimation (with discussion). J. Statist. Plann. Infer Groeneveld, R.A. and Meedan, G. (984). Measuring skewness and kurtosis. Statistician Hildebrand, D. K. (97). Kurtosis measures bimodality? The Amer. Statist

20 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL. Hogg, R.V. (972). More light on the kurtosis and related statistics.j. Amer. Statisti. Assoc Horn, P.S. (983). A measure for peakedness. The Amer. Statist James Kurian and Sebastian, G. (25). A comparative study between a class of mesokurtic distributions and normal model. STARS: Int. Journal Kale, B.K. and Sebastian, G. (996). On a class of non-normal symmetric distributions of kurtosis three. Statistical Theory and Applications, Paper in honor of H. A. David Ed. H. N. Nagaraja, P. K. Sen and P. R. Morrison, Springer-Verlag, New York, Lawrence, M.J. (975). Inequalities of s-ordered distributions. Ann. Statist Loh, W.Y. (982) Tail Orderings on Symmetric Distributions with Statistical Applications. Unpublished PhD thesis. University of Califorma. Berkeley. 6. MacGillivray, H.L. (986). Skewness and asymmetry: Measures and Orderings. Ann. Statist MacGillivray, H.L. and Balanda, K.P. (988a). Mixtures, myths and kurtosis. Comm. Statist. B Simulation Comput MacGillivray, H.L. and Balanda, K.P. (99). Kurtosis and spread. Canad. J. Statist Pearson, K.(95). Das fehlergeselz and seine verallgemeiner ungen durch fechner and Pearson. A Rejoinder. Biometrika, Rivest, L.P. (982). Product of random variables and star-shaped ordering. Canad. J. Statist

21 CHAPTER 3. A CLASS OF MESOKURTIC S AS ALTERNATIVE TO NORMAL 2. Ruppert D. (987). What is kurtosis?. An Influence function Approch. Ann. Statist Sebastian, G. and James Kurian (22). On Robustness of mean for non-normal symmetric distributions with kurtosis three. J. of Indian Statis. Assoc Van Zwet, W.R. (964). Convex Transformations of Random Variables. Math. Centre Tracts 7. Mathematisch Centrum. Amsterdam. 53