数据统计与分析基础实验三：常规数学统计计算
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- 1、随机生成一个10x15的高斯矩阵，均值为自己学号后两位，方差为1。对该矩阵分别进行LU、QR、奇异值，并展示分解结果。
-
- 2、利用软件求解优化问题：min Z = - X1 - 0.8X2 - 1.2X3, 约束：X1 –X2 + X3 <= 30, 3X1 + 2X2 + 4X3 <=42, 3X1 + 2X2 <= 30, X1, X2 , X3 >=0。并找到其中有效约束。
- 3、对于函数f(x) = (x2-5x+3)exp(-4x)cosx, x = 0: 0.1:1,进行三种方式插值，并将插值曲线与原曲线绘制在同一个Figure窗口。
- 4、统计自己过去12个月实际生活花费的数值，并拟合成一条一次、二次、三次曲线，三条曲线分别用不同颜色和线型展示在同一张图里。
- 5、求解三重积分∫（-0.5，1）∫（0，0.5）∫（-0.5pi,pi）(y sinx exp(x)+z cosx x2)dxdydz.
- 6、求微分方程 xy’+ y – exp(x) = 0,在初始条件y(1) = 2 exp下的特解，并画出解函数的图形。
- 7、 某人进行射击，及每次命中的概率为0.1，独立射击50次，求击中10次以上且40次以下的概率。
- 8、假设自己过去12个月实际生活花费的数值服从正态分布，请求出其均值和方差的极大似然估计。
1、随机生成一个10x15的高斯矩阵，均值为自己学号后两位，方差为1。对该矩阵分别进行LU、QR、奇异值，并展示分解结果。
LU
> X<-matrix(rnorm(150,79,1),nrow=10,ncol=15,byrow=T)
>
> X
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15]
[1,] 79.28764 79.28067 79.06171 77.86130 79.92227 78.97803 80.19302 78.09195 80.12844 79.33108 79.07045 81.65337 77.83014 79.16602 76.44480
[2,] 78.84768 78.64431 79.12125 78.62161 79.20040 78.80970 78.73790 79.90102 79.35878 79.00298 79.69092 79.28335 80.06084 76.60801 78.48222
[3,] 79.96133 79.44971 78.41336 77.65514 78.49294 77.61689 79.46074 78.41758 78.29983 78.08997 81.00809 78.36508 79.34868 78.78832 76.39192
[4,] 77.75511 79.42457 79.92943 79.68850 77.51813 79.90277 79.25886 78.71600 77.85925 79.97145 79.44708 79.05540 79.35358 80.32224 77.71266
[5,] 79.09960 77.77367 80.70951 80.78063 78.81128 79.11045 78.52318 79.50448 79.10117 78.30584 79.34353 77.81714 80.09034 78.44839 80.20928
[6,] 77.32727 78.23094 79.61353 80.92026 77.00957 77.93693 80.35393 80.52509 77.61259 80.49315 78.47708 77.24535 79.15108 78.19794 77.97894
[7,] 79.78001 78.23899 78.05903 81.42659 79.24841 78.22265 79.27916 78.83118 77.72174 77.34338 77.36072 79.11792 77.75593 79.01005 77.64143
[8,] 79.14777 79.58133 80.26682 79.91078 79.82839 79.38519 80.49733 77.77436 77.41137 78.69928 77.23665 80.30657 79.12233 80.41864 78.32603
[9,] 78.28875 78.58956 78.79957 78.26176 80.27090 78.26865 79.02158 79.30549 79.33774 78.56470 78.68895 79.66592 79.05017 80.39068 77.38571
[10,] 78.58806 78.09882 79.44993 80.73510 79.12248 78.46254 79.49634 80.48909 78.09370 78.71434 78.56228 79.03188 77.47336 78.16855 78.46260
> expand(lux<-lu(X))
$L
10 x 10 Matrix of class "dtrMatrix" (unitriangular)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 1.000000000 . . . . . . . . .
[2,] 0.972408865 1.000000000 . . . . . . . .
[3,] 0.989223152 -0.378332467 1.000000000 . . . . . . .
[4,] 0.997732293 -0.475577388 0.347079743 1.000000000 . . . . . .
[5,] 0.979082547 0.369980937 0.146700242 -0.031699053 1.000000000 . . . . .
[6,] 0.967058284 0.645343280 0.310732760 0.350660645 -0.179429136 1.000000000 . . . .
[7,] 0.991574779 0.230892682 0.101340905 -0.165944893 0.652181982 -0.445637838 1.000000000 . . .
[8,] 0.989825531 0.433776199 0.232757310 -0.013889100 0.452607506 0.183494824 0.201566422 1.000000000 . .
[9,] 0.982825740 0.006272751 0.520631403 0.374560254 0.280494778 0.301510971 0.065332617 -0.260755584 1.000000000 .
[10,] 0.986072576 0.138960838 0.284267601 -0.026876671 0.433385293 -0.299111830 0.199503762 -0.648373256 0.874030464 1.000000000
$U
10 x 15 Matrix of class "dgeMatrix"
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
[1,] 79.96133 79.44971 78.413357 77.655136 78.4929396 77.61689055 79.4607439 78.4175822 78.2998290 78.0899688 81.0080909 78.3650780 79.3486826
[2,] 0.00000 2.16696 3.679591 4.175960 1.1909035 4.42741917 1.9905305 2.4620458 1.7198043 4.0360731 0.6740984 2.8525048 2.1942145
[3,] 0.00000 0.00000 4.533309 5.542276 1.6148091 4.00506056 0.6718547 2.8634660 2.2958260 2.5844099 -0.5365188 1.3757839 2.4269316
[4,] 0.00000 0.00000 0.000000 4.009931 0.9393688 1.49727984 0.7120774 0.7684726 -0.3794612 0.4529684 -2.9568653 1.8096275 -1.2116282
[5,] 0.00000 0.00000 0.000000 0.000000 2.7721016 0.09716843 0.4105032 1.2215825 1.6906259 0.2501340 -0.8890861 1.7401994 0.1550007
[6,] 0.00000 0.00000 0.000000 0.000000 0.0000000 -1.73243541 1.8413765 2.1617961 0.5052528 1.4539308 0.7465506 -1.1289148 0.6988155
[7,] 0.00000 0.00000 0.000000 0.000000 0.0000000 0.00000000 1.5450996 -0.2293943 0.9181463 1.2651940 -0.9345461 1.8127753 -1.5933193
[8,] 0.00000 0.00000 0.000000 0.000000 0.0000000 0.00000000 0.0000000 -2.4724941 -2.4204228 -1.5771919 -2.7020287 0.2605034 -0.8297543
[9,] 0.00000 0.00000 0.000000 0.000000 0.0000000 0.00000000 0.0000000 0.0000000 -1.2429997 -0.5774546 -0.2911517 0.4024430 -1.7024896
[10,] 0.00000 0.00000 0.000000 0.000000 0.0000000 0.00000000 0.0000000 0.0000000 0.0000000 0.2734360 -0.9119530 -0.3654972 2.1996718
[,14] [,15]
[1,] 78.7883242 76.3919239
[2,] 3.7077790 3.4284784
[3,] 1.9119272 5.9377238
[4,] 1.5001369 0.9923843
[5,] 1.6456686 0.4836298
[6,] -1.2126053 -0.2153066
[7,] -1.3730681 -0.9435328
[8,] 0.1538485 -0.1333075
[9,] -0.8133704 -0.1457831
[10,] -2.0927327 0.9718066
$P
10 x 10 sparse Matrix of class "pMatrix"
[1,] . . . . . . | . . .
[2,] . . . . . . . . . |
[3,] | . . . . . . . . .
[4,] . | . . . . . . . .
[5,] . . | . . . . . . .
[6,] . . . . . | . . . .
[7,] . . . | . . . . . .
[8,] . . . . . . . | . .
[9,] . . . . | . . . . .
[10,] . . . . . . . . | .
>
>
QR
>
> qr(X)
$qr
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] -249.2264887 -248.9607965 -250.87989039 -251.65073888 -249.64198733 -248.7574082 -251.3293317 -250.28346791 -248.2074393 -249.3149401
[2,] 0.3163696 2.9132348 2.61145014 0.98187200 0.66391915 2.6653878 2.7370520 1.73838602 1.3303129 4.1686201
[3,] 0.3208380 0.1528378 3.27612404 3.73595046 0.38131165 1.9572461 0.8261335 2.73696991 1.0142605 2.0199458
[4,] 0.3119857 -0.5952122 -0.09557788 -3.73940084 0.54250670 -0.2069258 -1.0602672 -1.60591604 1.2522346 -0.6952599
[5,] 0.3173804 0.4326068 -0.72444665 -0.26124951 2.61357716 0.4594159 0.3103866 0.51819459 1.2394289 -0.0248041
[6,] 0.3102691 -0.3322246 -0.33575981 0.40149267 0.26088107 1.2910551 -1.0457994 -0.93361945 0.4090450 -0.3402466
[7,] 0.3201105 0.5062414 0.23564686 0.82493520 -0.34269746 0.2816665 -1.1664586 0.44868875 -0.5126468 -0.7833503
[8,] 0.3175737 -0.1713743 -0.10272400 -0.03793401 -0.05479495 0.1943184 -0.2907023 2.84942678 1.9459087 1.5591726
[9,] 0.3141269 -0.1255633 0.04600547 -0.06600955 -0.62805452 0.2372021 -0.8841201 0.45650549 -1.1222655 -0.4275769
[10,] 0.3153279 0.1455470 -0.27254469 0.26073528 -0.12473436 0.4306166 -0.1045288 0.03888371 -0.4923620 -0.1754419
[,11] [,12] [,13] [,14] [,15]
[1,] -249.4582707 -250.3066465 -249.5607478 -249.65236726 -246.3336239
[2,] 2.0351688 2.2365590 2.4204124 3.32299381 1.0048710
[3,] 0.5704595 -0.4497459 2.0937964 0.55917724 4.1518873
[4,] 1.2519630 0.5755954 0.6013972 -0.73328905 -0.6995292
[5,] -1.4514335 2.3415186 -0.6656064 1.21516467 0.3225719
[6,] -0.2158874 1.0951689 -0.2280997 -0.03147505 0.2286951
[7,] 0.8606732 -1.4187992 1.4747500 0.67411670 0.8499550
[8,] 2.4714592 -0.2054328 0.8740892 -1.75024932 0.5288189
[9,] -0.5756725 0.2379873 -0.7826350 -1.45217589 0.2017076
[10,] 0.5851269 0.2345101 -1.4113524 1.34273826 -0.6235301
$rank
[1] 10
$qraux
[1] 1.3181349 1.0473455 1.4591606 1.1274956 1.6336291 1.8007552 1.3505675 1.8888705 1.8703905 0.1754419 0.5851269 0.2345101 1.4113524 1.3427383
[15] 0.6235301
$pivot
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
attr(,"class")
[1] "qr"
> qr.Q(qr(X))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] -0.3181349 0.02662043 -0.25074081 0.34416189 0.15062377 0.2025038 -0.772087790 0.04147118 -0.18435474 0.14574560
[2,] -0.3163696 -0.04095622 -0.04351398 0.21132999 0.05754902 0.2494406 0.137117711 0.48787289 0.34300094 -0.64161956
[3,] -0.3208380 -0.14635828 -0.51775951 0.26901673 -0.55596151 -0.3713061 0.237859373 0.01943516 0.02362004 0.17528018
[4,] -0.3119857 0.60151295 0.02675873 -0.13009647 -0.26997008 0.5696713 0.239325984 -0.04644029 -0.06441149 0.24942782
[5,] -0.3173804 -0.42619719 0.67095765 0.31464218 -0.21570769 0.1136136 0.096211493 -0.12146693 -0.28870712 0.04051213
[6,] -0.3102691 0.33849067 0.27145230 -0.39961870 -0.21352456 -0.5134559 -0.282128549 0.22972943 -0.23105708 -0.24654914
[7,] -0.3201105 -0.49977655 -0.28848338 -0.65225153 0.05014704 0.2567901 0.005417173 -0.16316611 -0.17399061 -0.11711886
[8,] -0.3175737 0.17778791 0.03958669 0.08806974 0.14070941 -0.1637352 -0.035062654 -0.77036634 0.37964791 -0.26977852
[9,] -0.3141269 0.13190721 -0.10775810 0.13784300 0.66206685 -0.2238934 0.422849736 0.08067166 -0.41086808 0.10358143
[10,] -0.3153279 -0.13917880 0.21487796 -0.19160713 0.19812782 -0.1222875 -0.048076658 0.25219145 0.60307538 0.56079504
> qr.R(qr(X))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] -249.2265 -248.960796 -250.879890 -251.650739 -249.6419873 -248.7574082 -251.3293317 -250.2834679 -248.2074393 -249.3149401
[2,] 0.0000 2.913235 2.611450 0.981872 0.6639192 2.6653878 2.7370520 1.7383860 1.3303129 4.1686201
[3,] 0.0000 0.000000 3.276124 3.735950 0.3813116 1.9572461 0.8261335 2.7369699 1.0142605 2.0199458
[4,] 0.0000 0.000000 0.000000 -3.739401 0.5425067 -0.2069258 -1.0602672 -1.6059160 1.2522346 -0.6952599
[5,] 0.0000 0.000000 0.000000 0.000000 2.6135772 0.4594159 0.3103866 0.5181946 1.2394289 -0.0248041
[6,] 0.0000 0.000000 0.000000 0.000000 0.0000000 1.2910551 -1.0457994 -0.9336194 0.4090450 -0.3402466
[7,] 0.0000 0.000000 0.000000 0.000000 0.0000000 0.0000000 -1.1664586 0.4486888 -0.5126468 -0.7833503
[8,] 0.0000 0.000000 0.000000 0.000000 0.0000000 0.0000000 0.0000000 2.8494268 1.9459087 1.5591726
[9,] 0.0000 0.000000 0.000000 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 -1.1222655 -0.4275769
[10,] 0.0000 0.000000 0.000000 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 -0.1754419
[,11] [,12] [,13] [,14] [,15]
[1,] -249.4582707 -250.3066465 -249.5607478 -249.65236726 -246.3336239
[2,] 2.0351688 2.2365590 2.4204124 3.32299381 1.0048710
[3,] 0.5704595 -0.4497459 2.0937964 0.55917724 4.1518873
[4,] 1.2519630 0.5755954 0.6013972 -0.73328905 -0.6995292
[5,] -1.4514335 2.3415186 -0.6656064 1.21516467 0.3225719
[6,] -0.2158874 1.0951689 -0.2280997 -0.03147505 0.2286951
[7,] 0.8606732 -1.4187992 1.4747500 0.67411670 0.8499550
[8,] 2.4714592 -0.2054328 0.8740892 -1.75024932 0.5288189
[9,] -0.5756725 0.2379873 -0.7826350 -1.45217589 0.2017076
[10,] 0.5851269 0.2345101 -1.4113524 1.34273826 -0.6235301
>
奇异值
> svd(X)
$d
[1] 966.652468 6.713068 5.245613 4.380018 3.447728 3.198833 2.327248 1.845607 1.370596 1.218602
$u
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] -0.3168723 -0.57937235 0.002979980 -0.008681674 -0.37697765 0.26306893 0.232153894 -0.25872547 0.34427080 -0.336394490
[2,] -0.3163500 0.04150666 0.482878205 0.303703801 -0.08451119 0.31090003 0.344766011 0.24128901 -0.53646806 0.064106013
[3,] -0.3151237 -0.28318087 0.467198900 -0.093375225 0.21230184 -0.69448688 0.043880214 0.07650871 0.17781076 0.163499530
[4,] -0.3167697 0.06392744 -0.012046386 -0.615167748 0.22714076 0.12772421 -0.005032644 -0.55148555 -0.36940088 0.092025602
[5,] -0.3172203 0.40167816 0.139920732 0.327425310 0.52456230 0.22086891 -0.033700656 -0.20233882 0.43355336 -0.239263161
[6,] -0.3154828 0.51833140 0.061093030 -0.422739794 -0.43315545 -0.10163253 -0.032504854 0.35501995 0.15191560 -0.318174446
[7,] -0.3149368 0.04307530 -0.467427984 0.385969751 -0.05324131 -0.46212563 -0.001099071 -0.16315277 -0.39073682 -0.368520985
[8,] -0.3173059 -0.15035261 -0.534452333 -0.158820384 0.34718146 0.12884198 0.348073878 0.49557174 0.09839185 0.228658864
[9,] -0.3162259 -0.27470442 0.005496748 0.040070129 0.04117424 0.21455494 -0.835467646 0.25457048 -0.11103814 0.005650308
[10,] -0.3159801 0.21997739 -0.144676299 0.242528129 -0.40770889 -0.01428239 -0.062729940 -0.24668684 0.19968327 0.707758434
$v
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] -0.2578093 -0.173640210 -0.001620601 0.36262153 0.143726579 -0.50436129 0.28066180 -0.028986862 0.107725634 -0.12838301
[2,] -0.2575593 -0.194319425 0.012362407 -0.23722406 0.041589361 -0.20477910 0.21392978 0.104067972 -0.275214702 0.20928805
[3,] -0.2595626 0.177645080 -0.033756852 -0.13048871 0.210802919 0.28917137 0.17108880 -0.004265629 0.574853804 0.22521956
[4,] -0.2603538 0.506701763 -0.430105441 0.14234953 -0.107864937 -0.28237337 0.04785501 -0.256441673 -0.151989166 -0.28603931
[5,] -0.2582515 -0.290581367 -0.145057190 0.43571984 -0.007074256 0.11640779 -0.24627673 0.340318508 -0.006284875 0.24768988
[6,] -0.2573604 0.001315954 -0.093617830 -0.10206344 0.163252291 0.27637113 0.23346999 -0.397968939 -0.322793003 0.08392678
[7,] -0.2600149 -0.051481267 -0.156529915 -0.21859585 -0.307313194 -0.23600221 0.22261558 0.391383111 0.373639023 -0.07915543
[8,] -0.2589443 0.326835635 0.176119143 0.13836495 -0.459579067 -0.05897397 -0.38549196 0.097211193 -0.215145130 0.34615684
[9,] -0.2567793 -0.183283084 0.276107311 0.25357765 -0.163874198 0.27036382 -0.21729941 -0.198844987 0.212489812 -0.60909318
[10,] -0.2579539 0.107823095 0.116261322 -0.45191493 -0.371736360 0.20558543 0.06680208 0.057552815 -0.023965688 -0.15933395
[11,] -0.2580721 -0.064291840 0.596544130 -0.07282704 0.068127632 -0.32016806 -0.04726648 -0.396698286 0.123829490 0.25595053
[12,] -0.2589458 -0.450478062 -0.210355458 0.08839535 -0.232207427 0.27777863 0.23527099 -0.133219585 -0.215831710 0.08051379
[13,] -0.2581900 0.120715041 0.313154312 -0.09597329 0.432147547 0.04841573 0.01401577 0.499230173 -0.376854848 -0.33097692
[14,] -0.2582826 -0.207966050 -0.372559689 -0.37862193 0.314927689 -0.14222153 -0.64649306 -0.126270167 0.079583316 -0.03821155
[15,] -0.2548542 0.373856977 -0.040697299 0.27116466 0.282222108 0.27133392 0.05622709 0.050781973 0.113884408 0.18250772
>
2、利用软件求解优化问题：min Z = - X1 - 0.8X2 - 1.2X3, 约束：X1 –X2 + X3 <= 30, 3X1 + 2X2 + 4X3 <=42, 3X1 + 2X2 <= 30, X1, X2 , X3 >=0。并找到其中有效约束。
> obj<-c(-1,-0.8,-1.2)
> mat<-matrix(c(1,3,3,-1,2,2,1,4,0),nrow=3)
> dir<-c("<=","<=","<=")
> rhs<-c(30,42,30)
> max<-FALSE
> types<-c("I","I","I")
> Rglpk_solve_LP(obj,mat,dir,rhs,types=types,max = max)
$optimum
[1] -15.6
$solution
[1] 0 15 3
$status
[1] 0
$solution_dual
[1] NA
$auxiliary
$auxiliary$primal
[1] -12 42 30
$auxiliary$dual
[1] NA
$sensitivity_report
[1] NA
>
3、对于函数f(x) = (x2-5x+3)exp(-4x)cosx, x = 0: 0.1:1,进行三种方式插值，并将插值曲线与原曲线绘制在同一个Figure窗口。
4、统计自己过去12个月实际生活花费的数值，并拟合成一条一次、二次、三次曲线，三条曲线分别用不同颜色和线型展示在同一张图里。
money<-c(255,356,689,1245,1006,1165,547,897,1357,1024,975,651)
month<-c(1:12)
cost<-data.frame(month,money)
g4= ggplot(cost)+
geom_point(aes(x=month,y=money,color="point"))+
geom_smooth(aes(x=month,y=money,color="x"),alpha = 0,method='lm',span=1,formula = y~I(poly(x,1)))+
geom_smooth(aes(x=month,y=money,color="x^3"),alpha=0,method='lm',formula = y~I(poly(x,3)),span=1)+
geom_smooth(aes(x=month,y=money,color="x^2"),alpha=0,method='lm',formula = y~I(poly(x,2)),span=1)+
coord_cartesian(xlim = c(1,12),ylim=c(0,1500))+
scale_x_continuous(breaks = seq(1,12,by=1))+
scale_y_continuous(breaks = seq(0,1500,by=100))
g4
5、求解三重积分∫（-0.5，1）∫（0，0.5）∫（-0.5pi,pi）(y sinx exp(x)+z cosx x2)dxdydz.
> f<-function(x,y,z){
+ y*sin(x)*exp(x)+z*cos(x)*x^2
+ }
> arr<-array(1:1000)
> x1<--0.5*pi
> x2<-pi
> y1<-0
> y2<-0.5
> z1<--0.5
> z2<-1
> v<-(x2-x1)*(y2-y1)*(z2-z1)
> for(i in 1:1000){
+ r1<-runif(10^6,min = x1,max = x2)
+ r2<-runif(10^6,min = y1,max = y2)
+ r3<-runif(10^6,min = z1,max = z2)
+ h<-f(r1,r2,r3)
+ out<-v*mean(h)
+ arr[i]<-out;
+ }
> show<-mean(arr)
> show
[1] 1.098548
> show
[1] 1.098548
>
6、求微分方程 xy’+ y – exp(x) = 0,在初始条件y(1) = 2 exp下的特解，并画出解函数的图形。
7、 某人进行射击，及每次命中的概率为0.1，独立射击50次，求击中10次以上且40次以下的概率。
> show<-0
> for(x in 10:40){
+ temp<-dbinom(x,size = 50,prob = 0.1)
+ show=show+temp
+ print(paste("temp:",temp))
+ print(paste("show:",show))
+ }
[1] "temp: 0.0151833341172624"
[1] "show: 0.0151833341172624"
[1] "temp: 0.00613468045141914"
[1] "show: 0.0213180145686815"
[1] "temp: 0.00221530127412358"
[1] "show: 0.0235333158428051"
[1] "temp: 0.000719499559117061"
[1] "show: 0.0242528154019222"
[1] "temp: 0.00021128161656612"
[1] "show: 0.0244640970184883"
[1] "temp: 5.63417644176324e-05"
[1] "show: 0.0245204387829059"
[1] "temp: 1.36941788515078e-05"
[1] "show: 0.0245341329617574"
[1] "temp: 3.04315085589064e-06"
[1] "show: 0.0245371761126133"
[1] "temp: 6.19901100274016e-07"
[1] "show: 0.0245377960137136"
[1] "temp: 1.16004884261805e-07"
[1] "show: 0.0245379120185979"
[1] "temp: 1.99786189561996e-08"
[1] "show: 0.0245379319972168"
[1] "temp: 3.17120935812692e-09"
[1] "show: 0.0245379351684262"
[1] "temp: 4.6447005750344e-10"
[1] "show: 0.0245379356328962"
[1] "temp: 6.28268676816245e-11"
[1] "show: 0.0245379356957231"
[1] "temp: 7.85335846020313e-12"
[1] "show: 0.0245379357035765"
[1] "temp: 9.0749919984569e-13"
[1] "show: 0.024537935704484"
[1] "temp: 9.69550427185568e-14"
[1] "show: 0.0245379357045809"
[1] "temp: 9.57580668825249e-15"
[1] "show: 0.0245379357045905"
[1] "temp: 8.73982356467482e-16"
[1] "show: 0.0245379357045914"
[1] "temp: 7.36690108899799e-17"
[1] "show: 0.0245379357045914"
[1] "temp: 5.72981195810956e-18"
[1] "show: 0.0245379357045914"
[1] "temp: 4.1073920846663e-19"
[1] "show: 0.0245379357045914"
[1] "temp: 2.70973783363401e-20"
[1] "show: 0.0245379357045914"
[1] "temp: 1.64226535371762e-21"
[1] "show: 0.0245379357045914"
[1] "temp: 9.12369640954217e-23"
[1] "show: 0.0245379357045914"
[1] "temp: 4.6342584937357e-24"
[1] "show: 0.0245379357045914"
[1] "temp: 2.14549004339616e-25"
[1] "show: 0.0245379357045914"
[1] "temp: 9.02007826052439e-27"
[1] "show: 0.0245379357045914"
[1] "temp: 3.42868471891285e-28"
[1] "show: 0.0245379357045914"
[1] "temp: 1.17219990390182e-29"
[1] "show: 0.0245379357045914"
[1] "temp: 3.5817219285889e-31"
[1] "show: 0.0245379357045914"
> print(show)
[1] 0.02453794
8、假设自己过去12个月实际生活花费的数值服从正态分布，请求出其均值和方差的极大似然估计。
