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Консультация № 64615
28.11.2006, 18:34
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Ребята... стыдно, но "горю"... Вопрос по теории вероятности.
В куб (x1,x2,x3): 0<=xi<=1, i=1,2,3 наобум брошена точка.
Пусть (Y1,Y2,Y3) - ее координаты. Найти функцию распределения F(x)=P(Y1+Y2+Y3<x), плотность F‘(x). Построить их график.
Всем откливнувшимся - огромное спасибо.
В куб (x1,x2,x3): 0<=xi<=1, i=1,2,3 наобум брошена точка.
Пусть (Y1,Y2,Y3) - ее координаты. Найти функцию распределения F(x)=P(Y1+Y2+Y3<x), плотность F‘(x). Построить их график.
Всем откливнувшимся - огромное спасибо.
Обсуждение
Неизвестный
28.11.2006, 20:04
общий
это ответ
Здравствуйте, Китайник Владислав Сергевич!
В куб наобум кидается точка, то есть все три координаты у нее случайны в предалых 0..1 и независимы. Сумма случайных координат точки Y1+Y2+Y3, таким образом, также случайная величина в пределах 0..3, распределенная равномерно. Для равномерно распределенной функции график F(x) будет представлять собой отрезок между точками (0,0) и (1,3), а производная этой функции(плотность) - отрезок от (0,1/3) до (3,1/3).
P.S. Либо я не заметил какого-то подвоха, либо задача действительно крайне простая.
В куб наобум кидается точка, то есть все три координаты у нее случайны в предалых 0..1 и независимы. Сумма случайных координат точки Y1+Y2+Y3, таким образом, также случайная величина в пределах 0..3, распределенная равномерно. Для равномерно распределенной функции график F(x) будет представлять собой отрезок между точками (0,0) и (1,3), а производная этой функции(плотность) - отрезок от (0,1/3) до (3,1/3).
P.S. Либо я не заметил какого-то подвоха, либо задача действительно крайне простая.
Неизвестный
29.11.2006, 02:58
общий
Ай-яй-яй! Сумма очков, выпавших на двух кубиках у нас вовсе не равномерно распределена, хотя кол-во очков на одном — очень даже.
Неизвестный
29.11.2006, 03:04
общий
это ответ
Здравствуйте, Китайник Владислав Сергевич!
Геометрическая интерпретация задачи такова: вероятность равна объему пересечения полупространства (Y1+Y2+Y3<x) и куба (0≤x1,x2,x3≤1), деленному на объем всего куба. Поскольку объем единичного куба равен 1, то просто объем этого пересечения. Плотность вероятности — соответствено, площади этого сечения.
Функция имеет примерно такой вид
F(x) =
x<sup>3</sup>/6, если 0≤x≤1
(x<sup>3</sup> - 3(x-1)<sup>3</sup>)/6, если 1≤x≤2
(x<sup>3</sup> - 3(x-1)<sup>3</sup> + 3(x-2)<sup>3</sup>)/6, если 2≤x≤3
Геометрическая интерпретация задачи такова: вероятность равна объему пересечения полупространства (Y1+Y2+Y3<x) и куба (0≤x1,x2,x3≤1), деленному на объем всего куба. Поскольку объем единичного куба равен 1, то просто объем этого пересечения. Плотность вероятности — соответствено, площади этого сечения.
Функция имеет примерно такой вид
F(x) =
x<sup>3</sup>/6, если 0≤x≤1
(x<sup>3</sup> - 3(x-1)<sup>3</sup>)/6, если 1≤x≤2
(x<sup>3</sup> - 3(x-1)<sup>3</sup> + 3(x-2)<sup>3</sup>)/6, если 2≤x≤3
Форма ответа
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