Задание 2.

Задана функция распределения F(x,y) двумерной случайной величины (X,Y).

1. Найти вероятность попадания случайной точки (X,Y) в прямоугольник D, заданный неравенствами axb, cyd.

2. Найти двумерную плотность распределения случайной величины (X,Y).

3. Найти вероятность P{(X,Y)D}, используя двумерную плотность распределения.

1. F(x,y) = ,

a = , b = 0, c = –1, d = 1.

2. F(x,y) = ,

a = , b = 0, c = 0, d = .

3. F(x,y) = ,

a = –2, b = 2, c = –, d = 0.

4. F(x,y) = ,

a = –1, b = 0, c = – , d = 0.

5. F(x,y) = ,

a = , b = , c = 0, d = +.

6. F(x,y) =

a = –1, b = 2, c = 0, d = 4.

7. F(x,y) =

a = 2,5, b = 5, c = 0, d = 1.

8. F(x,y) =

a = 2, b = 4, c = 1, d = 6.

9. F(x,y) =

a = 2,5, b = 4, c = 2, d = 6.

10. F(x,y) =

a = 1, b = 3, c = 0, d = +.

11.F(x,y)=

a = –, b = 2, c = 1, d =2,5.

12. F(x,y) =

a = 1,5, b = 3, c = 5, d = 6.

13. F(x,y) =

a = -1, b = 1, c = 1, d = 4.

14. F(x,y) =

a = –, b = 0,5, c = 6, d = 9.

15. F(x,y) =

a = –1, b = 2, c = 0, d = 1.

16. F(x,y) = F₁(x)^.F₂(y), где F₁(x) =

F₂(y) = a = 2, b = 3, c = –2, d = 1.

17. F(x,y) = F₁(x)^.F₂(y), где F₁(x) =

F₂(y) = a = –1, b = –0,5, c = 0, d =1,5.

18. F(x,y) = F₁(x)^.F₂(y), где F₁(x) =

F₂(y) = a = –2, b = 0,5, c = –0,5, d = 1.

19. F(x,y) = F₁(x)^.F₂(y), где F₁(x) =

F₂(y) = a = 0,25, b = 1,25, c = –0,25, d = 1,25.

20. F(x,y) = F₁(x)^.F₂(y), где F₁(x) =

F₂(y) = a = 0,5, b = 1,5, c = 0,5, d = 4,5.

21. F(x,y) =

a = 1, b = 2, c = 0, d = 2.

22. F(x,y) =

a = 0, b = 1, c = –2, d = 1.

23. F(x,y) =

a = –2, b = 1, c = 2, d = 4.

24. F(x,y) =

a = 1, b = 2, c = 0, d = +.

25. F(x,y) =

a = –, b = 1, c = 0, d = 2.

26. F(x,y) =

a = 0, b = 2, c = –1, d = 1.

27. F(x,y) =

a = 1, b = 2, c = 0, d = +.

28. F(x,y) =

a = –3, b = , c = 1, d = 3.

29. F(x,y) =

a = –2, b = 1, c = 1, d = .

30. F(x,y) =

a = –1, b = , c = 0, d = +.

Задание 3.

Двумерная случайная величина (X,Y), распределенная в области G, задана плотностью распределения p(x,y).

1. Найти значение параметра а.

2. Найти вероятность попадания случайной точки (X,Y) в заданную область D.

3. Найти плотности распределения p₁(x) и p₂(y) случайных величин X и Y и условные плотности распределения p₁(xy) и p₂(yx). Сделать вывод о том, являются ли данные случайные величины зависимыми.

4. Найти M(X), M(Y), D(X), D(Y), K_XY.

1. p(x,y) = axy⁴, G = {(x,y): 0x2, 0y1};

D = {(x,y): x1, 0y }.

2. p(x,y) = ax²y, G = {(x,y): x1, 0y2};

D = {(x,y): x2, y1}.

3. p(x,y) = ax²y², G = {(x,y): 0x2, y1};

D = {(x,y): –2x1, 0y }.

4. p(x,y) = ax³y², G = {(x,y): 0x1, –2y0};

D = {(x,y): x2, y1}.

5. p(x,y) = axy³, G = {(x,y): 0x3, 0y1};

D = {(x,y): x1, 0y }.

6. p(x,y) = axsiny, G = {(x,y): 0x2, 0y };

D = {(x,y): 0x , 0y }.

7. p(x,y) = ay²cosx , G = {(x,y): – x , y1};

D = {(x,y): x , 0y2}.

8. p(x,y) = ax³cos , G = {(x,y): 0x2, yπ};

D = {(x,y): x1, – yπ}.

9. p(x,y) = ay²sin2x , G = {(x,y): 0x , y2};

D = {(x,y): x , –1y }.

10. p(x,y) = axsin , G = {(x,y): 0x2, 0y3π};

D = {(x,y): 1x2, π y3π}.

11. p(x,y) = asinxsiny, G = {(x,y): 0x , 0y };

D = {(x,y): – x , y }.

12. p(x,y) = acosxsiny, G = {(x,y): x , 0yπ};

D = {(x,y): – x0, 0y }.

13. p(x,y) = asin cosy, G = {(x,y): 0xπ, 0y };

D = {(x,y): 0x2π, 0y }.

14. p(x,y) = asin2xcos , G = {(x,y): 0x , 0yπ};

D = {(x,y): x , y }.

15. p(x,y) = asin cos , G = {(x,y): 0x π, y };

D = {(x,y): x , y }.

16. p(x,y) = a, G = {(x,y): 0x2, 0y2};

D = {(x,y): x0, y0, x+y2}.

17. p(x,y) = a, G = {(x,y): x1, 0y2};

D = {(x,y): 2xy}.

18. p(x,y) = a, G = {(x,y): 0x2, y1};

D = {(x,y): xy}.

19. p(x,y) = a, G = {(x,y): x0, y0, x+y2};

D = {(x,y): xy}.

20. p(x,y) = a, G = {(x,y): x2, y2};

D = {(x,y): xy}.

21. p(x,y) = a, G = {(x,y): x²+y²1};

D = {(x,y): yx}.

22. p(x,y) = a, G = {(x,y): x²+y²4};

D = {(x,y): x+y0}.

23. p(x,y) = a, G = {(x,y): y0, x²+y²1};

D = {(x,y): x0}.

24. p(x,y) = a, G = {(x,y): x0, x²+y²4};

D = {(x,y): x²+y²1}.

25. p(x,y) = a, G = {(x,y): y0, x²+y²9};

D = {(x,y): x²+y²4}.

26. p(x,y) = ae^–2x–y, G = {(x,y): x0, y0};

D = {(x,y): 0x2, y1}.

27. p(x,y) = ae^–x–3y, G = {(x,y): x0, y0};

D = {(x,y): –1x2, 0y2}.

28. p(x,y) = ae^–3x–2y, G = {(x,y): x0, y0};

D = {(x,y): x2, y1}.

29. p(x,y) = a^.2^–x–y, G = {(x,y): x0, y0};

D = {(x,y): –1x3, 0y2}.

30. p(x,y) = a^.3^–x–0,5y, G = {(x,y): x0, y0};

D = {(x,y): 0x2, –2y3}.