Handbook of mathematics for engineers and scienteists part 215 pot

1466 INDEX elliptic paraboloid, 145 elliptic point, 394 Emden–Fowler equation, 1223 empirical distribution function, 1084 variance, 1088 energy conservation law, 996 entire axis, 825, 826, 841 entire function, 414 entry, pivot, 1016 envelope, 377 of family of curves, 377 equal complex numbers, 399 equal matrices, 168 equal polynomials, 155 equation Abel differential, first kind, 462 Abel differential, first kind, canonical form, 463 Abel differential, second kind, 464 Abel differential, second kind, canonical form, 465 Abel functional, 914 Abel integral, 1385 Abel integral, generalized, 806, 1386 Abel integral, second kind, 1392 Abel integral, second kind, generalized, 1393 Airy, 955 algebraic, 162 algebraic, binomial, 161 algebraic, fourth-degree, 159 algebraic, linear, 157 algebraic, nth-degree, 162 algebraic, quadratic, 158 algebraic, reciprocal, 160, 161 algebraic, reciprocal, modified, 160 autonomous, 1207, 1223, 1225 axisymmetric steady hydrodynamic boundary layer, 712 Babbage, 917, 1416, 1423 Bernoulli, 1207 Bessel, 1216 Bessel, modified, 1216 biharmonic, 1297 biharmonic, homogeneous, 1297 biharmonic, nonhomogeneous, 1298 bilinear functional, 930, 1446 binomial algebraic, 161 biquadratic, 159 Born–Infeld, 659 Boussinesq, 709, 1332 Burgers, 664, 666, 751, 1307 Burgers, radial symmetric case, 1307 Burgers–Korteweg–de Vries, generalized, 709 Cantor first, 1442 Cantor second, 1449 Carleman, 1399 Cauchy, 1438 Cauchy exponential, 1443 Cauchy logarithmic, 1439 Cauchy power, 1444 equation (continued) characteristic, 514, 654 characteristic, for difference equation, 881 characteristic, for system of ODEs, 529 characteristic, of linear operator, 211 characteristic, of matrix, 185 characteristic, of quadratic form, 150 circle, canonical, 97 circle, parametric, 98 Clairaut, 572, 1262 complete, of plane, 124 continuity, 784 convolution-type, of first kind, 825 convolution-type, of second kind, 829 cubic, complete, 159 cubic, incomplete, 158 curve, on plane, 84 cyclic functional, 920 D’Alembert, 1441 damped vibrations, 1213 Darboux, 458 difference, see difference equation differential, see differential equation diffusion, linear, see heat equation diffusion, nonlinear, with cubic source, 753 diffusion, stationary anisotropic, 1323 diffusion boundary layer, 1276 dispersion, 532, 596 Duffing, 501 ellipse, canonical, 98 elliptic, 585, 588, 590, 1284 Emden–Fowler, 1223 Ermakov, 1224 Euler, 473, 994, 1216 Euler–Darboux, 602 Euler–Lagrange, 767 Euler–Poisson, 1006 evolution, second-order, 739 exact, first-order ODE, 458 exact, second-order ODE, 491 family, 377 Fisher, 1301 Fitzhugh–Nagumo, 1302 Fredholm, of fi rst kind, 825 Fredholm, of second kind, 829 free oscillations, 1213 functional, see functional equation functional differential, 682, 684, 688 gas dynamics model, 558 Gauss, 1445 Gauss, generalized, 1445 Gaussian hypergeometric, 960, 1218 Gel’fand–Levitan–Marchenko integral, 1328, 1333 general, of plane, 124 general, of second-order hypersurface, 220 general, of straight line in space, 132 general, of straight line on plane, 90 Hamilton–Jacobi, 576 heat, 585, 672, 680, 698, 728, 1267 heat, nonhomogeneous, 1268 heat, stationary, with nonlinear source, 1320 INDEX 1467 equation (continued) heat, stationary anisotropic, 1323 Helmholtz, 1289 Helmholtz, three-dimensional, 600 Helmholtz, two-dimensional, 599 homogeneous, 591, 592, 810, 829, 834, 1207, 1224 homogeneous, generalized, 1210, 1224 homogeneous function, 1441 hyperbola, canonical, 101 hyperbolic, 585, 587, 590 hyperbolic, first canonical form, 587 hyperbolic, normal, 590 hyperbolic, second canonical form, 587 hypergeometric, 960, 1218 hypergeometric, degenerate, 1215 incomplete, of plane, 124 information theory, basic, 1440 integral, see integral equation intercept, of plane, 125 intercept-intercept, of straight line, 91 involutory functions, 1423 Jacobi, 998, 1004, 1007 Jensen, 1438 Jensen, generalized, 1449 Khokhlov–Zabolotskaya, 658 Khokhlov–Zabolotskaya, stationary, 1322 Klein–Gordon, 618, 625, 1280 Klein–Gordon, linear, 625 Klein–Gordon, nonlinear, 702, 1315 Klein–Gordon, two-dimensional, 618 Kolmogorov, backward, 1076 Kolmogorov, forward, 1076 Korteweg–de Vries, 752, 756, 759, 1327 Korteweg–de Vries, cylindrical, 1328 Korteweg–de Vries, modified, 1328 L’Huiller, 74 Lalesco–Picard, 1403 Laplace, ODE, 520 Laplace, three-dimensional, PDE, 643 Laplace, two-dimensional, PDE, 585, 633, 1284 Legendre, 1217–1219 Lienard, 1225 light rays, 1259 likelihood, 1092 line, in projections, 133 linear, algebraic, 157 linear, constant-coefficient, ODE, 473, 514, 1213 linear, constant-coefficient, PDE, 588 linear, first-order, ODE, 457 linear, integral, 801, 810, 824, 829 linear, nonhomogeneous, ODE, 473 linear, ODE, 472, 518, 1213 linear, PDE, 585, 591, 592 Liouville, 664 Lobachevsky, 1444 logarithmic Cauchy, 1439 logistic, 875 Mathieu, 980, 1221 equation (continued) Mathieu, modified, 982, 1220 minimal surfaces, 770 model, of gas dynamics, 1254 Monge–Amp ` ere, 1326 Monge–Amp ` ere, homogeneous, 668, 1326 Monge–Amp ` ere, nonhomogeneous, 1326 Newell–Whitehead, 733 noncentral hypersurface, canonical, 224 normal, 252 normal to surface, 153 normalized, of plane, 126 normalized, of straight line, 91 on entire axis, 825, 826, 841 on semiaxis, 826, 842 one-dimensional vibration, with central symmetry, 1281 ordinary differential, see ordinary differential equation Painlev ´ e, 495 parabola, canonical, 104 parabolic, 585, 586, 590 parabolic, in narrow sense, 590 parametric, of curve on plane, 87 parametric, of line in space, 131 parametric, of plane, 125 partial differential, see partial differential equation pencil of planes, 131 pencil of straight lines, 92 Pexider, 925, 1440 Pfaffian, 772, 773 plane passing through line of intersection of planes, 131 plane passing through point and parallel to another plane, 127 plane passing through point and parallel to two straight lines, 129 plane passing through three points, 127 plane passing through two points and parallel to line, 128 Poisson, 1285 Poisson, three-dimensional, 599 Poisson, two-dimensional, 599 quadratic, 158 reciprocal, algebraic, 160, 161 reciprocal, algebraic, generalized, 160 reciprocal, algebraic, modified, 160 reciprocal, functional, 919 renewal, 1395 Riccati, 460 Riccati, general, differential, 461, 462, 1207, 1210 Riccati, special, differential, 1208 Riccati difference, 875, 918, 1428, 1429 right-hand side, 801, 825, 829 root, 162 Schl ¨ omilch, 1400 Schr ¨ oder–Koenigs, 911 Schr ¨ odinger, linear, 1276 Schr ¨ odinger, nonlinear, 759, 1309 Schr ¨ odinger, of general form, 1311 1468 INDEX equation (continued) Schr ¨ odinger, with cubic nonlinearity, 1309 Schr ¨ odinger, with power-law nonlinearity, 1310 self-similar solutions, 1442 separable, 456, 1207 separated, 456 sine-Gordon, 1314 sinh-Gordon, 1314 skew self-distributivity, 1449 slope-intercept, of straight line, 89 steady transonic gas flow, 1324 straight line in polar coordinates, 92 straight line in projections in space, 133 straight line in space, canonical, 132 straight line passing through given point on plane, canonical, 90 straight line passing through point and perpendicular to plane, 134 straight line passing through two given points, 90, 134 straight line passing through two given points on plane, canonical, 90 surface, 386 surface in space, 119 tangent line, 251 tangent plane, 268 tangent plane to surface, 152 telegraph, 1284 telegraph, generalized, 626 thermal boundary layer, 1276 transonic gas flow, 1324 transposed, 825, 829, 834 traveling wave, 1441 Tricomi integral, 1403 ultrahyperbolic, 590 Urysohn, with degenerate kernel, 863, 864 Urysohn integral, nonlinear, 856 vibration of string, 1278 vibration with axial symmetry, 1282 Volterra, with quadratic nonlinearity, 856 Volterra integral, nonlinear, 856, 858–861 Volterra integral, of fi rst kind, 801, 807 Volterra integral, of second kind, 807, 810 wave, 585, 1278 wave, nonlinear, 723, 1312–1317 wave, three-dimensional, 618 wave, two-dimensional, 618 Whittaker, 960 Wiener–Hopf, of first kind, 825, 1401 Wiener–Hopf, of second kind, 829, 1406 with difference kernel, integral, 825, 829 with quadratic nonlinearity, 865 with weak singularity of second kind, 829 equations admitting variational formulation, 767 autonomous, 525 Bellman, 1028 center of hypersurface, 222 difference, 871–907 elliptic, linear, 588, 609–611, 631–633, 637, 1284–1298 equations (continued) elliptic, nonlinear, 653, 654, 318–1334 Fredholm integral, of second kind, 854 functional, linear, 907–917 hyperbolic, linear, 587, 617, 623–630, 636, 647, 648, 1278–1284 hyperbolic, nonlinear, 653, 654, 1312–1317 integral, 801–871, 1385–1406 integral, linear, 801–855, 1385–1406 integral, linear, of second kind, with constant limits of integration, 824, 829 integral, nonlinear, 856–871 intrinsic, of curve, 388 motion of mass point, 87 ordinary differential, 453–550, 1207–1228 Painlev ´ e, 495 parabolic, linear, 586, 615–616, 618–622, 634, 635, 646, 648, 1267–1278 parabolic, nonlinear, 653, 654, 1301–1311 partial differential, 553–798, 1247–1382 pencil, 92 reaction-diffusion type, 796 straight line on plane parametric, 91 straight lines on plane, 89 equilateral hyperbola, 101 equilateral triangle, 43, 50, 51 equilibrium point of autonomous system, 545 equilibrium points of two-dimensional linear systems, 534 equilibrium solution, 874, 884 equivalence transformation, 180 equivalent events, 1031 equivalent infinitely small functions, 243 equivalent matrices, 180 equivalent representations of group, 230 Ermakov convergence criterion for series, 340 Ermakov equation, 1224 error function, xxxi, 939, 957, 1267 essential singularity, 413 estimation theorem for double integral, 318 for triple integral, 325 estimator, 1088 asymptotically unbiased, 1089 consistent, 1089 interval, 1088, 1093 maximum likelihood, 1092 minimality of variance, 1090 statistical, 1088 unbiased, 1089 Euclidean space, 192 isomorphic, 194 normed, 193 Euler constant, xxx, 944, 949, 958 Euler differential equations, 998 Euler equation, calculus of variations, 994 Euler equation, ODE, 473, 520, 1216 Euler formula, 30, 405, 944 Euler line, 47 Euler numbers, 939 Euler polynomials, 989 Euler substitutions, 280 INDEX 1469 Euler system of ODEs, 542 Euler theorem, 61, 393 Euler transform of first kind, 449 Euler transform of second kind, 450 Euler transformation, 661 Euler triangles, 71 Euler–Darboux equation, 602 Euler–Fourier formulas, 358 Euler–Lagrange equation, 767 Euler–Lagrange theorem, 1009 Euler–Poisson equation, 1006 even function, 236 even number, 3 even permutation, 178 event, 1031 complementary, 1031 elementary, 1031 favorable, 1031 frequency, 1083 impossible, 1031 random, 1031 relative frequency, 1083 sure, 1031 events difference, 1031 equivalent, 1031 incompatible, 1031 intersection, 1031 statistically independent, 1036 union, 1031 evolute, 378 evolutionary conditions, 792 evolvent, 378, 379 exact differential equation, 458 exact solutions of linear PDEs, 594 example Hilbert, 997 Weierstrass, 997 excenter, 47 excess, spherical, 72 excess coefficient, 1045 excircle of triangle, 47 excircles, 46 existence and uniqueness theorems for ODEs, 454, 488, 524, 543 for PDEs, 557, 577 expansion asymptotic, 363 asymptotic two-term, 478 of polynomial in powers of linear binomial, 156 expectation, 1042 conditional, 1066 of bivariate random variable, 1060 of continuous random variable, 1042 of discrete random variable, 1042 of function of multivariate random variable, 1065 population, 1087 expected value of random variable, 1042 exponent, H ¨ older, 310 exponential distribution, 1052 exponential form of complex number, 405 exponential function, 21 graphs, 21 properties, 22 exponential integral, 940, 958 exponential self-similar solution, 675 extended complex plane, 402 external angle, 43 of polygon, 51 extrema, relative, 991 extremal, 994, 998 admissible, 994, 998 broken, 997, 999 nonsingular, 995 regular, 995 regular point, 995 extremal points of function, 269 extremal properties of quadratic form, 220 extremum absolute, 991 first-order necessary condition, 994 necessary condition, 269 point, 269 sufficient conditions, 269 extrinsic curvature, 394 F face of dihedral angle, 60 of polyhedral angle, 60 of polyhedron, 61 of trihedral angle, 60 factor, reflection, 760 factor group, 227 factorial, xxxi, 10 double, xxxi, 10 faithful representation of group, 230 false negative rate, 1096 false position method, 262 false positive rate, 1095 family of curves, 377 of planes, 151 family equation, 377 favorable event, 1031 feasible region, 1012 feasible solution, 1012 Feller theorem, 1077 Ferrari solution, 161 Feuerbach points, 49 Feuerbach theorem, 49 fifth Painlev ´ e transcendents, 499 finite differences, 884 finite games, 1024 finite matrix, 167 finite series, 12 finite subgroup, 228 finite sums, 1111 finite-dimensional Euclidean space, 193 1470 INDEX first boundary value problem for elliptic equations, 633 for hyperbolic equations, 627 first boundary value problem, for ODEs, 480 for PDEs, 593 first comparison criterion, 338 first duality theorem, 1018 first group of derivation formulas, 396 first integral, 542, 544, 574 for system of ODEs, 544 first mean value theorem, 295 first noteworthy limit, 241 first Painlev ´ e transcendent, 496 first quadratic form of surface, 390 first variation of functional, 992 first-order differential constraint, 739 first-order hyperbolic system of quasilinear equations, 780 first-order linear difference equations, 1407 first-order necessary condition for extremum, 994 first-order neighborhood, 991 first-order ordinary differential equation Bernoulli, 458 linear, 457 first-order ordinary differential equations, 453– 472, 1207–1211 Cauchy problem, 454 homogeneous, 456 homogeneous, generalized, 457 integrable by quadrature, 453 not solved for derivative, 454 singular solutions, 455 solved for derivative, 453 uniqueness and existence theorems, 454 with separated or separable variables, 456 first-order partial differential equations, 553, 1247 Cauchy problem, 556 general solution, 553 linear, 1247 method of separation of variables, 571 nonlinear, 1256 quasilinear, 553, 1252 Fisher equation, 1301 Fisher information, 1089 Fisher test, 1104 Fisher transformation, 1105 Fisher–Snedecor distribution, 1086 Fitzhugh–Nagumo equation, 1302 fixed point of function, 908 flat plate coordinates of center of mass, 324 moments of inertia, 324 flux of vector field, 333 focal axis of ellipse, 99 of hyperbola, 101 focal distance of ellipse, 98 of hyperbola, 102 of parabola, 105 focal parameter of ellipse, 98 of hyperbola, 102 focal properties of hyperbola, 102 of parabola, 105 focal property of ellipse, 99 focal radius of parabola, 105 right, 99, 102 focus of hyperbola, 102 of parabola, 105 stable, for ODEs, 537 unstable, for ODEs, 537 focus-directrix property of ellipse, 100 of hyperbola, 103 of parabola, 105 Folium of Descartes, 374 form bilinear, 214, 992 bilinear degenerate, 215 bilinear nondegenerate, 215 bilinear skew-symmetric, 214 bilinear symmetric, 214 canonical, 462, 473 canonical for central surfaces, 143 canonical of gas dynamics systems, 786 canonical of matrix, 181 canonical of quadric, 148 canonical of second-order curve, 107 linear, 213 multilinear, 215 multilinear, skew-symmetric, 215 multilinear, symmetric, 215 normal, 473 quadratic, 216, 218, 840 quadratic, degenerate, 216 quadratic, negative definite, 216 quadratic, nondegenerate, 216 quadratic, nonnegative, 216 quadratic, nonpositive, 216 quadratic, positive definite, 216 quadratic, second of surface, 392 sesquilinear, 213 formal solution, 594 formula Bayes, 1036 Bernoulli, 1037 Bessel, 950 B ¨ urman–Lagrange, 356 Cauchy for ODE, 519 Cauchy integral, 409 Cauchy multiple integration, 289 Cauchy series multiplication, 343 Cauchy–Hadamard, 411 Chebyshev, 808 D’Alembert, 1278 de Moivre, 400 Dini, 429 Dirichlet, 295 INDEX 1471 formula (continued) Euler, 405, 944 for calculation of volume of domain, 327 Fourier cosine inversion, 445 Fourier inversion, 444 Fr ´ enet, 377 Gauss, 396, 808 Green, 428 Heron, 48 Hopf, 565 Kontorovich–Lebedev inversion, 448 Leibnitz, 256 Liouville, 473, 518 Liouville for system of ODEs, 540 Maclaurin, 257 Meijer inversion, 447 Newton–Leibniz, 288 Newton–Leibniz for function of complex variable, 408 of finite increments, 254 of integration by parts, 275, 409 of repeated integration by parts, 275 Peterson–Codazzi, 396 Poisson, 949, 1038 Post–Widder, 441 quadrature, 808 Serret–Fr ´ enet, 386 Stirling, 944 Sturges, 1083 total probability, 1036 Watson, 292 Willier, 74 formulas Bayes, 1063 D’Alembert (Gauss), 73 Euler–Fourier, 358 Fr ´ enet, 377 Frobenius, 174 Mollweide, 44 Peterson–Codazzi, 397 Serret–Fr ´ enet, 386 Weingarten, 396 forward Kolmogorov equation, 1076 Fourier coefficients, 358 Fourier cosine inversion formula, 445 Fourier cosine transform, 445, 449 asymmetric, 445 tables, 1177 Fourier expansion of even functions, 360 of odd functions, 360 Fourier inversion formula, 444 Fourier method, 602 Fourier series, 357, 358 criteria of uniform convergence, 361 differentiation, 359 Dirichlet–Jordan convergence criteria, 358 in complex form, 360 integration, 359 Lipschitz convergence criteria, 358 of square-integrable functions, 361 Riemann localization principle, 358 Fourier sine transform, 445, 446, 449 asymmetric, 446 tables, 1182 Fourier transform, 443, 449, 614, 843 alternative, 444 asymmetric, 444 asymmetric form, 444 inverse, 443 n-dimensional, 445 solving linear problems of mathematical physics, 615 standard form, 443 fourth Painlev ´ e transcendent, 498 fourth-degree algebraic equation, 159 fourth-order linear PDEs, 1294 fourth-order nonlinear PDEs, 1332 fraction, 6 fractional integral, 449 fractional power series, 415 Fredholm alternative, 834 for symmetric integral equations, 839 Fredholm integral equation first kind, 824, 825 second kind, 829, 852 second kind, with degenerate kernel, 830 second kind, with symmetric kernel, 835 Fredholm kernel, 825, 829 Fredholm theorems, 834 free oscillations, 1213 free terms of linear system of equations, 197 frequency cumulative, 1084 of event, 1083 of variate value, 1083 frequency histogram, 1086 frequency polygon, 1085 Fr ´ enet formulas, 377 Fresnel integrals, 304, 942 Frobenius formulas, 174 Frullani integral, 307 frustum of cone, 67, 68 of pyramid, 64 Fuchs index, 749 function admissible, 993, 1002, 1006 Airy, 477, 955 analytic, 413 analytic at point, 403 analytic at point at infinity, 403 analytic in domain, 403 antiperiodic, 885 approximation, 508 arccosine, xxx, 32 arccotangent, xxx, 32 arcsine, xxx, 31 arctangent, xxx, 32 autocorrelation, 1072 automorphic, 912 Bessel, 947, 1216 Bessel, modified, 953 1472 INDEX function (continued) Bessel, modified, of first kind, xxxi, 953, 1216 Bessel, modified, of second kind, xxxi, 953, 1216 Bessel, of first kind, xxxi Bessel, of second kind, xxxii beta, 945 boundary, 419 bounded, 236 bounded variation, 246, 247, 313 branch point, 404, 405 central cross-correlation, 1073 characteristic, 1046, 1066 Chebyshev, 985 complementary error, xxxi, 939, 958 complex variable, 401, 402, 407, 408 composite, 237, 265 concave, 245 continuous, 243 convex, 245 correlation, 1072 cosine, 24, 25 cosine, hyperbolic, xxx, 35 cotangent, xxx, 24, 26 cotangent, hyperbolic, xxx, 35 cross-correlation, central, 1073 cross-correlation, normalized, 1073 cross-covariance, 1072 cumulative distribution, 1039 cylindrical, 947, 1216 decreasing, 236 degenerate hypergeometric, xxxii, 956 differentiable, 251 differentiable at point, 402 digamma, 944, 949 Dirac delta, 599 distribution of two-dimensional random vector, 1057 distribution sample, 1084 domain, 236 elliptic, 972 entire, 414 entire transcendental, 414 error, xxxi, 939, 957, 1267 even, 236 exponential, 21 gamma, xxxii, 943, 1215 gamma, incomplete, xxxii, 946, 957, 958 Gauss error, xxxi Gegenbauer, 1218 generating, 815, 1047 generating, of numerical sequence, 1047 Green, 428, 631, 633–637 Hankel, 952 harmonic, 404 Heaviside unit step, 618 Hermite, 986 holomorphic at point, 403 homogeneous, 1441 Hurwitz, 893 hyperbolic arccosine, xxx function (continued) hyperbolic arccotangent, xxx hyperbolic arcsine, xxx hyperbolic arctangent, xxx hyperbolic cosine, xxx, 35 hyperbolic cotangent, xxx, 35 hyperbolic sine, xxxi, 35 hyperbolic tangent, xxxi, 35 hypergeometric, 960 incomplete beta, 947 incomplete gamma, xxxii, 946 increasing, 236 infinitely large, 243 infinitely small, 242 integrable, 286, 312, 314 integrand, 273 inverse, 237 inverse hyperbolic cosine, 40 inverse hyperbolic cotangent, 40 inverse hyperbolic functions, 39 inverse hyperbolic sine, 39 inverse hyperbolic tangent, 40 involutory, 917, 1423 Jacobi theta, 978 joint distribution, 1057, 1064 joint probability, 1059, 1064 Kummer, 956 Lagrange, 270 Legendre, 962 Legendre, associated, 964, 965 Legendre, associated, modified, 966 Legendre, associated, of first kind, 965 Legendre, associated, of second kind, 965 Legendre, of second kind, 1217 limit, 264 logarithmic, 22 Lyapunov, 549 Macdonald, 953 marginal distribution, 1064 marginal probability, 1059 Mathieu, 980, 1219 Mathieu, modified, 982 mean value, 318, 325 meromorphic, 414 moment-generating, 1047 monotone (monotonic), 236 multi-valued, 236, 402 multivariate random variable, 1065 Neumann, 947 nonincreasing, 236 normalized cross-correlation, 1073 odd, 236 of exponential growth, 905 of several variables, 263 of two variables, 263 Painlev ´ e transcendental, 495 parabolic cylinder, 967 penalty, 1027 periodic, 236 Pontryagin, 1011 power, 19 primitive, 273 . point of function, 908 flat plate coordinates of center of mass, 324 moments of inertia, 324 flux of vector field, 333 focal axis of ellipse, 99 of hyperbola, 101 focal distance of ellipse, 98 of hyperbola,. 102 focus of hyperbola, 102 of parabola, 105 stable, for ODEs, 537 unstable, for ODEs, 537 focus-directrix property of ellipse, 100 of hyperbola, 103 of parabola, 105 Folium of Descartes, 374 form bilinear,. variable, 1060 of continuous random variable, 1042 of discrete random variable, 1042 of function of multivariate random variable, 1065 population, 1087 expected value of random variable, 1042 exponent,