Question

In: Statistics and Probability

Suppose that X, Y and Z are normally distributed where X ≈ N(100,100), Y ≈ N(400,...

Suppose that X, Y and Z are normally distributed where X ≈ N(100,100), Y ≈ N(400, 400) and

Z ≈ N(64,64). Let W = X + Y + Z.

a) Describe the distribution of W, give a name and parameters E(W) and Var(W).

b) Use Excel or R to generate 200 random values for X, Y and Z. Add these to obtain 200 values for W. Create a histogram for W. In Excel use the NORMINV(rand(),mean, sd) function.

c) Estimate E(W) and Var(W) using the random numbers.

Solutions

Expert Solution

Here,
X follow N(100,100)
Y follow N(400,400)
Z follow N(64,64)
and W = X + Y + Z
(a) -> So the W follw N(100+400+64,100+400+64) = N(564,564)
Mean of W = E[W] = 564
Var[W] = 564

(b)->

S.N Random Values W = X + Y + Z
X Y Z
1 102.9725 429.7546 -78.7694 453.9576
2 216.6945 -275.791 46.79389 -12.3025
3 -45.822 -88.3059 100.7276 -33.4003
4 -21.1631 694.7918 41.39185 715.0205
5 -20.7497 549.178 -43.9027 484.5256
6 40.44401 630.6301 22.0071 693.0812
7 17.10229 -34.0736 202.4504 185.4791
8 269.1069 119.0052 115.5844 503.6965
9 37.9778 395.7618 96.76145 530.501
10 131.1535 475.4908 84.47862 691.1229
11 16.51839 643.469 73.96166 733.9491
12 89.1196 791.4474 -29.2693 851.2977
13 83.79963 -156.528 78.35932 5.630953
14 124.4809 206.2913 43.31001 374.0822
15 57.84594 37.32625 -23.411 71.76117
16 73.64648 1112.598 -40.4525 1145.792
17 -82.8694 650.3552 53.99381 621.4797
18 20.01199 709.1964 -2.08971 727.1186
19 44.15771 1114.099 103.8367 1262.093
20 -44.8761 659.9941 31.23294 646.3509
21 133.7338 -232.477 95.91046 -2.83287
22 198.481 672.2882 74.97884 945.748
23 17.84421 357.0632 32.89092 407.7983
24 173.187 240.1063 200.9161 614.2094
25 158.4635 -262.643 87.59579 -16.584
26 107.477 74.5828 195.7373 377.7971
27 200.9462 194.9963 69.28627 465.2287
28 261.6618 532.7344 -0.83402 793.5621
29 14.0764 193.4947 -43.4592 164.1119
30 14.90336 248.3918 116.3672 379.6624
31 10.90135 342.1343 98.05759 451.0933
32 -97.8133 530.0541 117.1167 549.3575
33 22.75252 600.8965 104.3551 728.0041
34 135.5697 35.70826 96.44379 267.7217
35 91.01851 436.971 93.33564 621.3251
36 249.2754 879.8421 -15.1813 1113.936
37 131.6277 349.0215 34.59364 515.2428
38 303.4794 853.2621 98.90102 1255.643
39 284.8593 1008.921 30.20725 1323.988
40 49.16743 87.61722 29.76749 166.5521
41 96.7597 -288.091 59.32272 -132.009
42 109.1812 699.2665 62.98406 871.4318
43 117.3175 768.1535 -4.73277 880.7381
44 110.7881 887.7911 61.55877 1060.138
45 -47.7542 683.3103 115.5099 751.0659
46 140.1807 173.4049 114.7956 428.3811
47 61.85943 609.7286 121.6815 793.2695
48 109.7191 674.9948 148.0143 932.7282
49 166.8332 302.9587 139.8478 609.6397
50 108.2598 301.6345 44.13882 454.0331
51 36.13119 915.3943 86.63943 1038.165
52 64.34883 297.5927 -54.8274 307.1141
53 -25.9723 1693.607 75.61039 1743.245
54 116.1151 65.81345 74.31367 256.2422
55 8.031668 104.3242 155.9309 268.2867
56 140.355 738.8077 119.51 998.6727
57 196.014 997.1942 128.8586 1322.067
58 34.18669 -217.139 54.96406 -127.989
59 53.45661 334.7647 99.3677 487.589
60 -47.3904 453.1081 84.3395 490.0572
61 128.9207 313.6089 68.94736 511.477
62 21.64453 896.0693 224.0524 1141.766
63 69.76067 553.5827 -0.23172 623.1117
64 86.96217 9.539373 13.60599 110.1075
65 132.0783 592.8493 24.35331 749.2809
66 40.76347 -4.09031 32.40517 69.07833
67 51.03019 354.1078 56.43046 461.5685
68 -17.7284 395.7008 -17.4296 360.5428
69 192.2255 822.8859 110.3483 1125.46
70 31.7929 1083.962 162.1669 1277.922
71 -1.43117 645.0543 55.9816 699.6047
72 -89.2586 781.9741 111.1475 803.863
73 141.5362 735.403 150.2801 1027.219
74 66.98602 1268.196 92.73327 1427.916
75 194.079 1034.622 135.1161 1363.817
76 -157.336 538.5033 14.49555 395.6628
77 82.36399 -585.034 105.8291 -396.841
78 298.7964 503.9571 113.0567 915.8102
79 37.03864 -326.963 29.71096 -260.213
80 256.9288 434.4207 -40.1571 651.1924
81 190.047 722.1057 123.99 1036.143
82 71.68504 787.6612 190.3246 1049.671
83 201.048 449.9299 123.77 774.7479
84 9.158532 421.108 104.5284 534.7949
85 222.4321 681.1121 -39.3372 864.2069
86 125.5702 461.9502 0.268578 587.789
87 91.14902 46.50851 69.65968 207.3172
88 215.1964 1068.384 17.80429 1301.385
89 -3.46093 1197.852 51.15808 1245.55
90 172.2106 608.3957 13.2112 793.8175
91 23.08199 311.0073 92.69536 426.7847
92 125.0172 242.3245 83.40956 450.7513
93 182.4887 596.4313 114.4541 893.3741
94 -126.322 522.3675 129.7818 525.827
95 -60.0165 158.0758 109.8027 207.862
96 -115.542 417.5542 130.3992 432.4113
97 208.7667 -315.761 -1.23609 -108.231
98 137.3684 537.8539 132.9072 808.1296
99 48.98451 -68.6572 56.32197 36.64932
100 207.4222 -160.094 43.32551 90.65363
101 76.01537 582.2268 69.14381 727.386
102 28.06715 710.9735 101.5153 840.556
103 113.1766 239.476 41.89659 394.5492
104 -113.705 830.452 94.06076 810.8079
105 65.99081 583.0422 130.5503 779.5833
106 136.4676 641.851 150.4627 928.7812
107 144.5239 733.9264 106.3336 984.7839
108 -171.677 90.06058 64.4578 -17.1586
109 43.23446 546.0667 -64.2344 525.0667
110 100.4552 -256.331 139.7491 -16.1271
111 98.72625 -162.304 4.915732 -58.6622
112 124.686 853.9024 114.0743 1092.663
113 102.628 334.5164 87.334 524.4784
114 19.56884 598.6436 -28.6509 589.5616
115 331.2445 197.0526 -1.63248 526.6646
116 243.214 10.76992 -21.6047 232.3793
117 -41.7779 313.9532 42.96412 315.1393
118 248.3031 803.3773 144.363 1196.043
119 65.53652 447.1864 112.2722 624.9951
120 173.127 304.5654 85.3462 563.0386
121 225.5671 511.6364 69.44343 806.6469
122 139.7913 -138.796 -39.6085 -38.6127
123 -151.322 478.1715 45.12296 371.972
124 73.63068 107.9713 64.86904 246.471
125 74.42183 623.0117 175.3408 872.7743
126 205.6412 -392.774 -1.23609 -188.369
127 136.4266 -120.918 43.39769 58.90668
128 144.5068 484.0123 45.07174 673.5908
129 58.9304 198.3741 79.53664 336.8412
130 169.2146 45.28615 202.6041 417.1048
131 141.9535 -178.109 207.2444 171.0885
132 164.2068 160.8256 17.73437 342.7668
133 205.4141 138.7577 162.04 506.2118
134 169.0204 462.4768 41.77704 673.2743
135 299.4067 281.2827 43.6605 624.35
136 -52.2058 657.166 27.38214 632.3424
137 90.78034 519.1329 113.4516 723.3649
138 -94.8183 320.9863 207.1245 433.2925
139 56.73136 265.3566 96.28794 418.3759
140 -25.6515 1018.957 128.0699 1121.375
141 10.44524 -420.077 40.3885 -369.243
142 196.0745 767.5932 0.413078 964.0808
143 98.3055 411.0322 67.33806 576.6757
144 75.88941 1352.859 3.522095 1432.271
145 70.82398 950.3125 41.28235 1062.419
146 122.0757 396.0678 126.0415 644.185
147 10.12487 813.0088 45.67958 868.8133
148 69.52829 444.9991 -44.1469 470.3804
149 110.0497 478.5458 153.4329 742.0283
150 13.35471 365.2719 128.3452 506.9718
151 20.55859 191.1837 93.61329 305.3555
152 121.2221 324.0417 -5.96561 439.2983
153 116.526 177.8459 -109.487 184.8851
154 145.854 -96.4659 108.1978 157.5859
155 143.8658 456.2895 99.43049 699.5859
156 161.8184 488.8049 159.0319 809.6551
157 210.68 503.6092 41.14673 755.436
158 144.8197 1560.282 19.05961 1724.162
159 127.766 594.3281 21.30628 743.4003
160 38.35752 947.5667 26.71697 1012.641
161 169.8185 -143.533 56.12465 82.41001
162 174.7864 180.3052 41.86537 396.9569
163 -9.7394 -42.2127 26.1819 -25.7702
164 170.2194 849.7906 26.56025 1046.57
165 82.20085 374.5396 56.28748 513.0279
166 39.1971 960.0123 98.27209 1097.481
167 41.01859 109.049 14.56161 164.6292
168 215.3451 465.018 82.29496 762.658
169 -34.0761 496.285 129.2444 591.4533
170 61.76075 295.2512 99.44759 456.4596
171 117.0457 959.1992 68.63799 1144.883
172 42.11646 130.7514 -62.4276 110.4402
173 90.35776 -391.253 85.04622 -215.849
174 23.51309 411.9203 12.13057 447.564
175 164.6869 714.5879 115.5369 994.8117
176 3.937578 701.0914 -28.3171 676.7119
177 40.39831 794.6207 91.90177 926.9207
178 36.6089 588.3586 55.67558 680.6431
179 86.18262 -33.578 10.41898 63.02364
180 167.5618 -358.693 126.45 -64.6814
181 245.2486 287.9821 19.84563 553.0763
182 147.4831 132.8969 80.91689 361.2969
183 67.91371 100.045 95.29098 263.2497
184 95.19787 989.4708 93.90782 1178.576
185 256.9028 404.7889 31.14359 692.8354
186 164.1974 435.3116 -3.3969 596.112
187 163.1574 133.8518 51.8816 348.8908
188 42.32439 1041.834 123.914 1208.072
189 79.13767 -526.317 56.34664 -390.832
190 -63.3234 804.8043 184.7175 926.1983
191 88.35744 287.9184 -31.9859 344.29
192 116.2313 633.9625 114.4474 864.6412
193 -35.3651 1650.533 17.42223 1632.591
194 -106.095 223.2552 98.88392 216.044
195 -139.004 301.067 55.51754 217.5802
196 52.85054 754.4419 44.59044 851.8829
197 5.885261 652.9287 98.23818 757.0521
198 233.5516 871.4038 -15.0447 1089.911
199 57.31025 -215.435 -45.5611 -203.686
200 -54.7896 -158.144 61.28439 -151.649
Mean 87.70363 425.8008 67.14061 580.645
Variance 9518.317 167807.1 3885.671 176657.3

and corresponding histogram of W is given below:


(c)->

From the last row of the above table,

E[W] = 580.645

Var[W] = 176657.3

orchestra answered 2 years ago
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