mean solution概率論與隨機(jī)過程

（（,（,1 ??的期望與方差（1（2）

???,???

0???1,?????? ??????????????????,????? f

??

?

0????x??

? ???

0?????

0???

??? ??

?????,?????2??1???0

? ? EX? ?fXxdx???dx ?

? ? EY??fY

dy??

dy??2??

dy?3?3?3?1 ? ? EX?Y?E

?EY?2?1?

3???

?? XXvar?X??

1?f?x?

??x?

dx?

?? ? varY?

fY

dy

y

y??2??

dy??y??

EXY?

?

xypx,ydydx????????? cov?X,Y??E?XY??E?X?E?Y?? 22有些同學(xué)在計(jì)算中使用var?X?Y??var?X??var?Y?，這個(gè)設(shè) 1

,??

??

????

,??0,??

?????????

???????,?????

???

??????E?Y??

????????,???

??????

?????E?XY??

?????????,??????

?????? Cov?X,Y??E?XY??E?X?E?Y??求??????|?|參考解

???????1?設(shè)Z?min?|X|1?，則0?Z?1，且有：pZ?0??p?z?，P?Z??1?Z在?0,1?pZ?z??p?z??p??z?? 0?????? ??1?

X?1??π?1?z?? ?2π?1?z??dz??π?1?z????? ??E?Z??????x?dx????????x?dx? ??x?dx? ?? 設(shè)隨 量與的相關(guān)系數(shù)為ρ，求U其中,b,c,d均為常數(shù)。E?U??aE?X??b，Var?U??E?V??cE?Y??d，Var?V??Cov?U,V??E?UV??

?與V

???E?acXY?adX?bcY?bd??acE?X?E?Y??adE?X??bcE?Y???acE?XY??ac?0時(shí)，Cov?U?0ac0時(shí)可求U、

?ac?Cov?X,

acρ ac?ρUV??0,ac? ac?為

???,

??

?

???

????! ?

??

試求解?與

?

?

?

?11?

??

??!?m????! E?????m? var?????m???1????var?N??N???var?????var??則? m???????1??????m???1???? ??m???令X???Z，其中，Z為非負(fù) 量且Z?（2）證明：Var????Var????Var??（3）利用（2）的結(jié)論，證明Var????Var?max?0,X???Var?max?0,因?yàn)閅Z?0，則E?YZ??Cov??,Z??E?YZ??E?Y?E?Z???E?Y?E?Z??

Var????Var????Var????2Cov??,Z??Var????Var??令U?max?0X?，V?max?0?當(dāng)X?0時(shí)，U?X，V?0此時(shí)UV?0，U?V?X，U、V當(dāng)X?0時(shí)，U?0，V??X此時(shí)UV?0，U??X，U、UV?0，U?V?X，U、V

Var????Var?max?0,X???Var?max?0,?當(dāng)X?0時(shí)，max?0X??Xmax?0?X??因此Var????Var?max?0,X???Var?max?0,?因此Var????Var?max?0,X???Var?max?0,?YZ?00YZ00時(shí)，討論就相當(dāng)于分段討論。而方差是對隨機(jī)等概率出現(xiàn)，則P?N?n?可以求解：A?A??50?n?P?N?n??5??46?1???46?2??…?n?50?49?…??50????5? 1??…? 4???46??n?3???…?n??50?49?…?46??1??…? ?5??49???…??46?50?…???

??…????CC選擇5個(gè)陽性患者的位置，共有C?種可能；對于符合N?n的序列，前n

?n?1?C???n?

? *

!

???1?

?1??!

??

?

??n?a?C???n?1?C???a?1?C?????1???????a? ? 式，得到組合數(shù)遞推： ?根據(jù)上述遞 C??C????C?? C?C????????

? ?? ??? ?

? N

?變量代換：k?45???

?

?

?45?k?

?

k?5 ? ? ?

?? ?

1

??n?1?C??

?1???

? ??

? ?

5 ? ????

? ??

?

n??

50

C?????? C C

CC

?

??1?50?51?6?解二：5個(gè)陽性患者隔出6個(gè)區(qū)間，編號1-6。設(shè)?為第? 患者所處區(qū)間??

??2????????

???6?6,? ?

?1???

???

? ??其中?????1?表 ????1?的示性變量第k個(gè)人拿到自己帽子的記為A?。 P

EA?

??1 ?0??1? ? P?A?

?1? N?N?E?AA??1

?0??1 ??? N?N? N?N? N?N?A? ?1

N?N? E?A????E?A????2 E?A?A???N? ??,?

? ??2 NN?2E?A?

?E?A??N

1? ?var?A??E?A???E?A????,??4?先求?X,Y?

??,5p?x,y??λ?λ?e????對第一象限y?x E?Z??? ??4x?1?p?x,y?dydx?? ?5yp?x,y?dxdy ???P?X?Y??4?xp?x,y?dydx?5? 由過去的習(xí)題知P?X?Y???? 代入E?Z?

? ?? λ?? min?

?

???????1????·??且達(dá)到最小值時(shí)?b?????????

??????????????記T????E??Y??aX??E?Y???2aE?XY??2bE?Y??a?E?X???