Problemas

1.- Genere, mediante el método de la Transformada Inversa, 100 nímeros aleatori

P(x)= 1/5 (4/5)x-1 nº

Ri 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Xi 0.4659 (1/5)] 0 .3 0 1 4 0 .0 1 5 8 0 .2 2 9 9 0 .3 5 6 9 0 .3 7 9 8 0 .9 6 5 4 0 .3 7 2 4 0 .0 6 3 2 0 .9 2 4 6 0 .0 4 8 6 0 .9 2 3 8 0 .0 6 8 4 0 .5 5 4 0 0 .8 7 5 2 0 .4 2 5 1 0 .1 5 2 2 0 .2 2 9 9 0 .4 7 0 9 0 .8 8 8 6 0 .2 0 9 8 0 .5 0 8 9 0 .9 5 6 7 0 .4 1 6 0 0 .9 8 2 7 0 .9 8 8 5 0 .3 8 2 6 0 .4 3 3 5 0 .7 6 5 4 0 .7 8 2 3 0 .4 1 8 3 0 .7 3 6 3 0 .7 7 6 1 0 .4 1 2 8 0 .2 5 7 1 0 .5 6 8 7 0 .3 9 7 2 0 .4 7 7 7 0 .9 1 8 6 0 .6 6 1 8 0 .8 2 9 2 0 .9 9 5 6 0 .3 3 2 2 0 .3 9 0 7

para x= 1,2,3,…

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

0 .7 3 7 2 0 .7 1 6 3 0 .1 5 7 4 0 .1 2 8 2 0 .2 1 6 0 0 .9 3 8 9 0 .5 0 8 1 0 .8 8 6 5 0 .8 5 2 2 0 .9 6 8 2 0 .1 3 4 7 0 .3 9 5 3 0 .0 6 8 2 0 .7 8 6 7 0 .1 7 2 9 0 .8 3 0 7 0 .7 5 0 9 0 .7 2 8 7 0 .9 0 7 5 0 .3 1 6 6 0 .4 5 6 6 0 .5 0 4 8 0 .1 9 8 7 0 .0 4 7 6 0 .1 1 7 9 0 .6 3 0 1 0 .9 6 6 5 0 .0 3 5 3 0 .4 7 7 1 0 .3 2 7 2 0 .5 4 0 3 0 .4 3 5 2 0 .1 9 2 2 0 .9 4 2 6 0 .0 2 0 4 0 .7 6 9 1 0 .7 7 9 9 0 .2 0 7 6 0 .0 0 5 8 0 .1 4 8 9 0 .5 8 9 7 0 .5 2 3 0 0 .9 1 2 3 0 .2 8 2 9 0 .4 0 6 9 0 .9 4 2 2 0 .0 5 1 5 0 .0 1 3 1 0 .0 7 7 9 0 .3 9 0 9

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

0 .7 3 7 2 0 .7 1 6 3 0 .1 5 7 4 0 .1 2 8 2 0 .2 1 6 0 0 .9 3 8 9 0 .5 0 8 1 0 .8 8 6 5 0 .8 5 2 2 0 .9 6 8 2 0 .1 3 4 7 0 .3 9 5 3 0 .0 6 8 2 0 .7 8 6 7 0 .1 7 2 9 0 .8 3 0 7 0 .7 5 0 9 0 .7 2 8 7 0 .9 0 7 5 0 .3 1 6 6 0 .4 5 6 6 0 .5 0 4 8 0 .1 9 8 7 0 .0 4 7 6 0 .1 1 7 9 0 .6 3 0 1 0 .9 6 6 5 0 .0 3 5 3 0 .4 7 7 1 0 .3 2 7 2 0 .5 4 0 3 0 .4 3 5 2 0 .1 9 2 2 0 .9 4 2 6 0 .0 2 0 4 0 .7 6 9 1 0 .7 7 9 9 0 .2 0 7 6 0 .0 0 5 8 0 .1 4 8 9 0 .5 8 9 7 0 .5 2 3 0 0 .9 1 2 3 0 .2 8 2 9 0 .4 0 6 9 0 .9 4 2 2 0 .0 5 1 5 0 .0 1 3 1 0 .0 7 7 9 0 .3 9 0 9

95 96 97 98 99 100

0 .9 7 6 7 0 .9 9 6 9 0 .3 1 7 4 0 .6 5 1 8 0 .6 1 6 9 0 .7 1 9 7

os para la siguiente distribución de probabilidad.

2.- Obtenga, con el metodo de la transformada inversa, la expresion matemática para gene

rar variables aleatorias que sigan las funciones de densidad indicadas

Genere con el método de la transformada inversa, variables aleatorias con densidad f(x) usand

lso numeros aleatorios 0.642, 0.225, 0.134, 0.176, 0.954, y 0.245

Obtenga el valor de la variable que se generaría mediante el método de composición al utilizar los n

meros aleatorios 0.5623 y 0.9825 con la funcion de densidad siguiente:

Geenre 50 variables aleatorias para las siguientes distribuciones de probabilid

ad; utilice el método de la transformada inversa.

Utilizando una hoja de cálculo, genere 50 variables aleatorias. a b c

triangularmente distribuidas con límite inferior = 5, valor más probable = 10 y límite su con distribución binomial y parametros N = 5, p = 0.3, q = 0.7 con distribución de Poisson, con = 3

compruebe con Staff: fit si las variables aleatorias generadas siguen la distribución de probabilidad qu a

a= Ri

n° 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

5 b= Xi 0.0902 0.7924 0.7481 0.0514 0.6770 0.5269 0.7385 0.7956 0.0956 0.8065 0.3954 0.1291 0.7237 0.9583 0.5538 0.2654 0.6157 0.6820 0.3023 0.1225 0.7217 0.9049 0.5061 0.2881 0.9362 0.9949 0.4010 0.9685 0.3172 0.9419 0.9362 0.4481 0.3290 0.3265 0.3759 0.8641 0.9617

7.1237 11.7780 11.4510 6.6034 10.9811 10.1365 11.3839 11.8032 7.1868 11.8899 9.4461 7.5410 11.2829 13.5566 10.2765 8.6425 10.6167 11.0127 8.8876 7.4744 11.2698 12.8189 10.0308 8.7952 13.2139 14.4939 9.4775 13.7441 8.9825 13.2960 13.2135 9.7335 9.0558 9.0403 9.3354 12.3932 13.6153

15 c=

38 39 40 41 42 43 44 45 46 47 48 49 50

b

0.0915 0.5437 0.0948 0.7622 0.6623 0.0328 0.4286 0.7797 0.5827 0.8654 0.9357 0.0612 0.6371

p= RI

n° 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

7.1386 10.2232 7.1777 11.5515 10.8911 6.2811 9.6294 11.6810 10.4324 12.4059 13.2071 6.7493 10.7404

xi 0.7328 0.2186 0.6383 0.4459 0.8679 0.4573 0.3039 0.6358 0.2879 0.4233 0.5865 0.1963 0.5639 0.9880 0.0158 0.3962 0.5791 0.0106 0.9036 0.9691 0.6809 0.7373 0.6685 0.8364 0.3774 0.8372 0.4555 0.5785 0.8022 0.1940

0.3 q= piezas x lote 1 0 2 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0

1

2

2

2

0.7

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

c

0.7638 0.4012 0.7163 0.1718 0.4335 0.6093 0.6301 0.2137 0.0775 0.8870 0.9403 0.9825 0.7348 0.3471 0.2758 0.2980 0.8090 0.1065 0.2337 0.9318

1 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1

3

λ=

n°

ri 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Xi 0.0217 0.9875 0.7734 0.9467 0.7740 0.0424 0.8472 0.3921 0.8418 0.5326 0.8111 0.0706 0.7877 0.5594 0.5043 0.5732 0.5046 0.6223 0.5017 0.4363 0.0124 0.7539 0.8724

2

1

3

2

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.7767 0.7101 0.7316 0.5913 0.7994 0.6825 0.5564 0.2662 0.1620 0.2137 0.5331 0.0239 0.3475 0.1876 0.5196 0.0817 0.3283 0.4139 0.4987 0.1293 0.3127 0.7054 0.3181 0.1884 0.0761 0.0320 0.9451

erior = 15

e se esperaría de ellas 10

genere una muestra de tamaño 50, de una distribución e

nº 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Ri 0.0559 0.4102 0.3184 0.5245 0.7331 0.5362 0.8217 0.0754 0.9088 0.2538 0.2521 0.5012 0.2090 0.0875 0.5004 0.2355 0.5974 0.3415 0.5837 0.6307 0.6715 0.5788 0.2684 0.2846 0.9547 0.7160 0.6327 0.0516 0.1983 0.5740 0.0021 0.3193 0.1426 0.7117 0.4437 0.3576 0.3217 0.0347 0.3749 0.5353 0.5781 0.9611 0.0687

Xi 0.1150 1.0560 0.7668 1.4869 2.6420 1.5366 3.4490 0.1568 4.7900 0.5856 0.5811 1.3909 0.4689 0.1831 1.3878 0.5371 1.8197 0.8355 1.7527 1.9925 2.2262 1.7294 0.6251 0.6698 6.1899 2.5176 2.0029 0.1060 0.4420 1.7066 0.0042 0.7694 0.3076 2.4874 1.1731 0.8850 0.7764 0.0706 0.9398 1.5327 1.7261 6.4917 0.1424

44 45 46 47 48 49 50

0.8028 0.1091 0.0829 0.8167 0.3412 0.4491 0.9560

3.2475 0.2311 0.1730 3.3931 0.8348 1.1925 6.2464

xponencial de parametro 2

genere una muestra de tamaño 50 con media = (los ultimos 4 digi

dni

47246959

media

6959

N°

R1

R2

R3

R4

R5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

0.451 0.079 0.798 0.846 0.982 0.578 0.655 0.170 0.463 0.554 0.274 0.573 0.918 0.563 0.003 0.360 0.837 0.590 0.513 0.933 0.645 0.618 0.417 0.066 0.949 0.223 0.932 0.420 0.937 0.093 0.673 0.204 0.675 0.314 0.303 0.622 0.028 0.775 0.626

0.884 0.375 0.898 0.007 0.705 0.904 0.809 0.655 0.004 0.765 0.330 0.502 0.211 0.956 0.522 0.761 0.710 0.725 0.934 0.585 0.079 0.659 0.617 0.566 0.462 0.905 0.246 0.965 0.658 0.521 0.232 0.396 0.165 0.176 0.507 0.567 0.857 0.852 0.841

0.186 0.728 0.449 0.290 0.396 0.328 0.174 0.762 0.464 0.459 0.155 0.889 0.186 0.383 0.806 0.008 0.505 0.611 0.996 0.551 0.280 0.403 0.166 0.209 0.600 0.174 0.432 0.135 0.643 0.073 0.122 0.275 0.172 0.802 0.054 0.415 0.054 0.215 0.999

0.342 0.531 0.990 0.442 0.030 0.901 0.389 0.347 0.011 0.820 0.566 0.218 0.279 0.757 0.162 0.180 0.601 0.317 0.861 0.727 0.486 0.114 0.062 0.265 0.355 0.253 0.026 0.628 0.966 0.108 0.962 0.093 0.989 0.633 0.028 0.547 0.826 0.567 0.203

0.938 0.890 0.630 0.121 0.484 0.214 0.127 0.507 0.510 0.447 0.320 0.677 0.049 0.698 0.164 0.356 0.694 0.006 0.410 0.078 0.667 0.790 0.315 0.692 0.895 0.911 0.893 0.277 0.673 0.035 0.781 0.497 0.737 0.262 0.494 0.945 0.339 0.009 0.774

40 41 42 43 44 45 46 47 48 49 50

0.305 0.080 0.181 0.182 0.245 0.472 0.556 0.103 0.302 0.117 0.583

0.086 0.468 0.911 0.313 0.630 0.977 0.790 0.000 0.734 0.437 0.314

0.750 0.225 0.834 0.788 0.542 0.311 0.813 0.570 0.805 0.299 0.927

0.129 0.183 0.374 0.613 0.307 0.118 0.336 0.293 0.232 0.094 0.534

0.028 0.613 0.007 0.837 0.909 0.834 0.368 0.445 0.916 0.018 0.852

os de tu dni) y desviacion estandar 2

R6

R7

R8

R9

R10

R11

R12

0.298 0.877 0.036 0.484 0.207 0.494 0.423 0.138 0.435 0.102 0.432 0.042 0.842 0.774 0.147 0.751 0.139 0.454 0.668 0.528 0.906 0.769 0.881 0.429 0.508 0.938 0.616 0.526 0.601 0.584 0.815 0.753 0.335 0.564 0.406 0.901 0.067 0.656 0.962

0.261 0.344 0.101 0.331 0.712 0.893 0.537 0.861 0.201 0.227 0.414 0.333 0.500 0.415 0.792 0.791 0.203 0.591 0.390 0.304 0.894 0.483 0.708 0.691 0.727 0.352 0.209 0.583 0.430 0.892 0.656 0.910 0.784 0.113 0.192 0.473 0.624 0.320 0.824

0.354 0.952 0.554 0.824 0.552 0.245 0.570 0.705 0.784 0.570 0.834 0.428 0.023 0.191 0.240 0.484 0.616 0.159 0.312 0.675 0.429 0.236 0.128 0.404 0.152 0.529 0.259 0.581 0.119 0.099 0.904 0.442 0.785 0.764 0.540 0.704 0.531 0.204 0.621

0.467 0.435 0.613 0.522 0.680 0.411 0.529 0.906 0.357 0.873 0.300 0.014 0.260 0.306 0.432 0.268 0.269 0.134 0.694 0.288 0.951 0.570 0.211 0.839 0.862 0.006 0.951 0.058 0.311 0.544 0.675 0.165 0.446 0.798 0.039 0.652 0.524 0.380 0.851

0.962 0.972 0.597 0.102 0.964 0.004 0.252 0.912 0.024 0.669 0.564 0.108 0.830 0.565 0.985 0.477 0.288 0.214 0.804 0.257 0.807 0.899 0.442 0.130 0.014 0.979 0.586 0.305 0.313 0.040 0.242 0.849 0.701 0.364 0.687 0.709 0.122 0.478 0.935

0.916 0.095 0.405 0.061 0.189 0.680 0.596 0.118 0.038 0.252 0.552 0.191 0.550 0.542 0.360 0.225 0.399 0.428 0.744 0.307 0.682 0.550 0.829 0.594 0.724 0.433 0.032 0.566 0.389 0.911 0.724 0.213 0.647 0.940 0.504 0.238 0.984 0.100 0.112

0.421 0.886 0.558 0.742 0.026 0.577 0.281 0.494 0.148 0.738 0.317 0.477 0.721 0.254 0.898 0.983 0.800 0.636 0.411 0.648 0.594 0.967 0.922 0.024 0.651 0.094 0.671 0.639 0.025 0.234 0.170 0.062 0.215 0.118 0.847 0.533 0.643 0.397 0.824

0.515 0.604 0.420 0.180 0.968 0.584 0.406 0.533 0.432 0.038 0.357

0.822 0.124 0.299 0.143 0.153 0.562 0.857 0.449 0.812 0.549 0.462

0.222 0.548 0.188 0.087 0.637 0.208 0.806 0.263 0.934 0.727 0.662

0.504 0.089 0.026 0.803 0.691 0.702 0.591 0.612 0.018 0.004 0.470

0.902 0.881 0.373 0.952 0.888 0.630 0.287 0.947 0.574 0.788 0.626

0.907 0.009 0.105 0.393 0.631 0.021 0.045 0.977 0.373 0.479 0.799

0.596 0.410 0.441 0.943 0.262 0.599 0.833 0.280 0.452 0.639 0.733

Suma Ri

Xi

6.479 7.165 6.629 4.772 5.927 6.227 5.341 6.575 3.437 6.478 5.059 4.451 5.369 6.404 5.511 5.646 6.060 4.865 7.736 5.883 7.421 7.058 5.698 4.908 6.900 5.797 5.852 5.682 6.066 4.135 6.957 4.860 6.652 5.850 4.600 7.305 5.598 4.953 8.572

6959.958 6961.329 6960.258 6956.543 6958.854 6959.455 6957.682 6960.150 6953.874 6959.955 6957.118 6955.902 6957.737 6959.808 6958.022 6958.292 6959.121 6956.731 6962.473 6958.766 6961.842 6961.116 6958.396 6956.817 6960.800 6958.593 6958.704 6958.363 6959.131 6955.270 6960.913 6956.719 6960.304 6958.700 6956.200 6961.610 6958.197 6956.905 6964.144

5.764 4.233 4.158 6.234 6.864 6.019 6.687 5.472 6.586 4.190 7.319

6958.527 6955.466 6955.316 6959.467 6960.728 6959.038 6960.375 6957.944 6960.172 6955.380 6961.639

para una distribución exponencial, genere 50 numeros sabiedno que α = 1.2 N°

ri

Xi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

0.454 0.567 0.502 0.053 0.117 0.871 0.535 0.761 0.986 0.505 0.731 0.326 0.150 0.736 0.653 0.623 0.882 1.000 0.692 0.713 0.045 0.229 0.737 0.110 0.643 0.716 0.017 0.591 0.196 0.061 0.179 0.500 0.030 0.992 0.036 0.360 0.612 0.051 0.771 0.331 0.035

0.726 1.004 0.836 0.065 0.149 2.460 0.920 1.718 5.082 0.844 1.574 0.473 0.195 1.598 1.269 1.172 2.560 10.014 1.413 1.499 0.056 0.312 1.602 0.140 1.236 1.509 0.021 1.073 0.262 0.075 0.236 0.832 0.037 5.767 0.044 0.536 1.135 0.063 1.768 0.483 0.043

42 43 44 45 46 47 48 49 50

0.065 0.591 0.990 0.022 0.122 0.268 0.540 0.340 0.193

0.080 1.074 5.584 0.027 0.157 0.375 0.932 0.498 0.257

sea X una variable aleatoria continua con densidad f. Hallar la densidad de la variable aleatoria Y = X

Use el metodo de la trasnformada inversa para generar una variable aleatoria con función de distrib 

  =

 + 2

, 0≤≤1