习题三龙永红版答案(2)

fY(y)??ba?b?a1dx???DD??0a?x?b,?(x)?y??(x)其它

1f(x,y)D当a?x?b,?(x)?y??(x)时,fxy(x|y)??b?afY(y)?D1 b?a1f(x,y)D?fxy(x|y)??b?afY(y)故fxy(x|y)为均匀分布

?1??(x)?y??(x),a?x?b ??b?a其它D??0又 fX(x)????(x)(x)??(x)??(x)1dy???DD?0?a?x?b其它

fyx11?f(x,y),a?x?b,?(x)?y??(x)?D ?????(x)??(x)fX(x)?(x)??(x)?0?D故 fyx(yx)为均匀分布.

17. 解： f?x,y??fy|xyxfx?x? fy?y?????????f?x,y?dx

故fxy?x,y??fyx?yx?f?x??????f?x,y?dx

18. f?x,y??fX?x??fY?y? 故独立. g?x,y??gX?x?gY?y? 不独立. 19. f?x,y??fX?x?fY?y? 不独立. 20. f?x????10?x?1

0其它?2?1?y?e2 f?y???2??0????y???,0?x?1其它

21. 若?x,y?服从二元正态分布，则

f?x,y??12????21??2?x?u1?22?12?e21??2?1???x?u?2(x?u1)(y?u2)?y?u2?2??2????22???2??12??1??1f?x??e2??1

f?y???1e2??2?y?u2?22?22

因为 f?x?,f?y?均无参数ρ，故可见，f?x,y?不能由fX?x?,fY?y?决定.

22. 解：

fX?x?????????1e2?x2?2?x2?y22?1?sinxsiny?dy

?1?sinxsiny?dy?1e2??y22

1e2??x22????ey2?21e2?x?2?????ey2?2dy1e2??,fY?y?? ???y???

fX?x?,fY?y?均服从正态分布，但是f?x,y?不服从正态分布.

23. ? ０ １ ２ ３ ４ P 1/9 4/9 4/9 0 0 ? 0 1 2 -1 -2 P 3/9 2/9 1/9 2/9 1/9 ? ? -2 -1 0 1 2 ０ １ ２ ３ ４ p????i? ０ 0 1/9 0 0 1/9 0 1/9 0 1/9 0 2/9 1/9 0 1/9 0 1/9 3/9 0 1/9 0 1/9 0 2/9 0 0 1/9 0 0 1/9 1/9 2/9 3/9 2/9 1/9 1 p????j? 24.

? ? 25.证明：

2 3 4 5 6 7 8 9 10 11 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 当n=2时，P?X1＋X2＝k???P?Xi?0kk1?i,X2?k?i? ?i,??p?X2?k?i?

??P?Xi?01 ??i?0ki?1i!??e??1??i?k2(k?i)!?ik?i?1?2

?e?(?1??2)?i?ok

i!(k?i)!(?1??2)k?(?1??2) ? ，k=0，1，2，?? ek!由数学归纳法，设对n-1成立，即

?Xi?1n?1i服从参数为

??的泊松分布，因为X服从参数为

in

n?1i?1?n的泊松分布，故?Xi?Xn服从参数为??i??n的泊松分布，即对n成立.

i?1i?1n?1n?126.

??e??xi,xi?0 i=1,2. ,Z=X1+ X2 fi(xi)??0x?0,i?Z＞0时，

f(Z)??f1(z?t)f2(t)dt???e??(z?t)??e??tdt

00zz ??e22??z?z0dt

??ze??z

所以fZ(z)?fx?x(z)??120,z?0 2??z??ze,z?0?X1?X2X－X２，Y=1，且X和Y服从正态分布 2227.由题意知X1~N(4,3), X2~N(2,1).且两者相互独立. 由X1=X+Y, X2=X-Y得X=

故EX?EX1?EX24?2??3

221111DX?DX1?DX2??3??1?1

4444EX1?EX24?211EY???1 , DY?DX1?DX2?1

2244故X~N(3,1),Y~N(1,1)

1?即fX(x)?e2?(x?3)221? , fY(y)?e2?(y?1)22

28. 用数学归纳法进行推广，与25题类似. Xi~N(?i,?i2),i=1,2??n.

?xi?1ni~N(??i,??i2)

i?1i?1nn1??,(x,y)?G29.f(x,y)??2

??0(x,y)?G当0＜z?2时，

FZ(z)?P??X??z????f(x,y)dxdy ?Y?G ??0???1???zy1??dxdy?dxdy ????0???2????zy2???? ?(0??11zy01dx)dy?0 21zydy 02z ?

4??当z＞2时，FZ(z)?1?1 z???0?0z?0???1?1故FZ(z)??z0?z?2 fZ(z)???4?4?1?1?1z?2???z?z230.

P?XY?0??0.3?0?0.3?0.2?0.8

z?00?z?2 z?2XY 0 P -1 1

0.8 0.1 0.1 P?XY??1??0.1 P?XY?1??0.1

X 0 1 ?0?0.8?(?1)?0.1?1?0.1?0 EXYE(X?Y)?EX?EY

P 0.6 0.4 Y -1 0 1 =0×0.6+0.4×1+(-1) ×0.4+0×0.2+1×0.4 P 0.4 0.2 0.4 =0.4 31 由题意知 f,X(x)???11?x?2?0其它 fY(y)???1?0所以:

f(x,y)???11?x?2,1?y?2?0其它 E?max(X,Y)????????????max(x,y)?1dxdy

??221?1max(x,y)dxdy

??221?21xdxdy??21?1ydxdy

x>y y>x ??2(?x2211xdy)dx??1(?xydy)dx

??221(x2?x)dx??2112(4?x)dx ?x32 2x2x32231?21?2x1?61

?83?13?32?2?76 ?53

32.证明： P(X=ai,Y=

j)=Pij, i,j=1,2…….

P(X= ai)＝Pi?

P(Y=bj)=P?j j=1,2……

??E(X?Y)???(ai?bj)?Pij

i?1j?1?y?2,其它1

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