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2,5 puntos
Parte obligatoria
La velocidad de crecimiento de la población de peces en un lago al cabo del tiempo, tt en años, viene dada por P′(t)=20 t e−t10P'(t) = 20\,t\,e^{-\frac{t}{10}}, siendo P(t)P(t) la población de peces. Sabiendo que la población inicial es de 10001000 peces, calcula cuántos tendrá el lago transcurridos 1010 años.

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FUNCIÓN DE DISTRIBUCIÓN NORMAL N(0,1)N(0,1)
Función de distribución normal N(0,1) Curva normal con el área P(Z ≤ z) sombreada desde menos infinito hasta z, a la derecha de cero. N(0,1) P(Z ≤ z) 0 z −∞ +∞
zz0,000,010,020,030,040,050,060,070,080,09
0,00,50000,50400,50800,51200,51600,51990,52390,52790,53190,5359
0,10,53980,54380,54780,55170,55570,55960,56360,56750,57140,5753
0,20,57930,58320,58710,59100,59480,59870,60260,60640,61030,6141
0,30,61790,62170,62550,62930,63310,63680,64060,64430,64800,6517
0,40,65540,65910,66280,66640,67000,67360,67720,68080,68440,6879
0,50,69150,69500,69850,70190,70540,70880,71230,71570,71900,7224
0,60,72570,72910,73240,73570,73890,74220,74540,74860,75170,7549
0,70,75800,76110,76420,76730,77040,77340,77640,77940,78230,7852
0,80,78810,79100,79390,79670,79950,80230,80510,80780,81060,8133
0,90,81590,81860,82120,82380,82640,82890,83150,83400,83650,8389
1,00,84130,84380,84610,84850,85080,85310,85540,85770,85990,8621
1,10,86430,86650,86860,87080,87290,87490,87700,87900,88100,8830
1,20,88490,88690,88880,89070,89250,89440,89620,89800,89970,9015
1,30,90320,90490,90660,90820,90990,91150,91310,91470,91620,9177
1,40,91920,92070,92220,92360,92510,92650,92790,92920,93060,9319
1,50,93320,93450,93570,93700,93820,93940,94060,94180,94290,9441
1,60,94520,94630,94740,94840,94950,95050,95150,95250,95350,9545
1,70,95540,95640,95730,95820,95910,95990,96080,96160,96250,9633
1,80,96410,96490,96560,96640,96710,96780,96860,96930,96990,9706
1,90,97130,97190,97260,97320,97380,97440,97500,97560,97610,9767
2,00,97720,97780,97830,97880,97930,97980,98030,98080,98120,9817
2,10,98210,98260,98300,98340,98380,98420,98460,98500,98540,9857
2,20,98610,98640,98680,98710,98750,98780,98810,98840,98870,9890
2,30,98930,98960,98980,99010,99040,99060,99090,99110,99130,9916
2,40,99180,99200,99220,99250,99270,99290,99310,99320,99340,9936
2,50,99380,99400,99410,99430,99450,99460,99480,99490,99510,9952
2,60,99530,99550,99560,99570,99590,99600,99610,99620,99630,9964
2,70,996530,996640,996740,996830,996930,997020,997110,997200,997280,99736
2,80,997440,997520,997600,997670,997740,997810,997880,997950,998010,99807
2,90,998130,998190,998250,998310,998360,998410,998460,998510,998560,99861
3,00,998650,998690,998740,998780,998820,998860,998890,998930,998960,99900
3,10,999030,999060,999100,999130,999160,999180,999210,999240,999260,99929
3,20,999310,999340,999360,999380,999400,999420,999440,999460,999480,99950
3,30,999520,999530,999550,999570,999580,999600,999610,999620,999640,99965
3,40,999660,999680,999690,999700,999710,999720,999730,999740,999750,99976
3,50,999770,999780,999780,999790,999800,999810,999810,999820,999830,99983
3,60,999840,999850,999850,999860,999860,999870,999870,999880,999880,99989
3,70,999890,999900,999900,999900,999910,999910,999920,999920,999920,99992
3,80,999930,999930,999930,999940,999940,999940,999940,999950,999950,99995
3,90,999950,999950,999960,999960,999960,999960,999960,999960,999970,99997
4,00,999970,999970,999970,999970,999970,999970,999980,999980,999980,99998
Nota: En el interior de la tabla se da la probabilidad de que la variable aleatoria ZZ, con distribución N(0,1)N(0,1), esté por debajo del valor zz.