1(3) P(Z3?0)?P(X?Y?0)?,
61117P(Z3?1)?P(X?0,Y?1)?P(X?1,Y?1)?P(X?1,Y?0)????,
312612111P(Z3?2)?P(X?0,Y?2)?P(X?1,Y?2)???.
12642? 设随机变量(X? Y)的联合分布律是
X ?1 1 Y ?1 1 求函数Z?X/Y的分布律? P(Z?X/Y?1)?P(X?Y?1)?P(X?Y??1)?0.25?0.25?0.5, P(Z?X/Y??1)?1?P(Z?X/Y?1)?0.5.
?2e?2x3? 设X与Y相互独立? 概率密度分别为fX(x)???0?e?yx?0f(y)??x?0,Y?0y?0
x?0,试求Z?X?Y的概率密度?
fZ(z)??f(x,z?x)dx??fX(x)fY(z?x)dx
00zz??2e?2xe?z?xdx?2e?z?e?xdx?2e?z(1?e?z),z?0.00zz
★4? 设X~U(0? 1)? Y~E(1)? 且X与Y独立? 求函数Z?X?Y的密度函数?
?e?y,0?x?1,y?0, f(x,y)???0,其它,当0?z?1时?
fZ(z)??f(x,z?x)dx??fX(x)fY(z?x)dx00zz??e?z?xdx?e?z?x0zzx?0?1?e?z,
当z?1时?
fZ(z)??f(x,z?x)dx??fX(x)fY(z?x)dx??e0001z1?z?xdx?e?z?x1x?0?e1?z?e?z.
因此
?1?e?z,0?z?1,?1?z?zfZ(z)??e?e,z?1,
?0,其它.??e?(x?y)★5? 设随机变量(X? Y)的概率密度为f(x,y)???1?e?1??00?x?1,0?y???其它
(1)求边缘概率密度fX (x)? fY (y)? (2)求函数U?max (X, Y)的分布函数? (3)求函数V?min (X,
Y)的分布函数?
?e?x?y?e,y?0,,0?x?1,?(1) fX(x)??1?e?1 fY(y)???0,其它.?0,其它.??0,x?0,?0,x?0,?x?x?xxe1?e???min{x,1}(2) FX(x)???fX(x)dx?? dx?,0?x?1,??1?e?1?1001?e1?e,x?0.???1?1?e1,x?1.???0,y?0, FY(y)???y?1?e,y?0.?(1?e?x)2,0?x?1,??1 FU(x)?FX(x)FY(x)??1?e?1?e?x,x?1.?(1?e?x)(1?e?min{x,1})?,x?0. ?11?e?1,x?0,??x?1?e?e(3) SX(x)@1?FX(x)??,0?x?1, ?1?1?e??0,x?1.?1,x?0,? ??e?min{x,1}?e?1,x?0.??1?1?e?1,y?0,SY(y)@1?FY(y)???y
?e,y?0.?(e?x?e?1)e?x1?e?1?e?2x?e?1?x?,0?x?1,?1??1?1 FV(x)?1?SX(x)SY(x)??1?e1?e?1,x?1.?1?e?1?e?min{x,1}?x?e?1?x?,x?0. ?11?e6? 设某种型号的电子管的寿命(以小时计)近似地服从N(160? 202)分布? 随机地选取4只求其中没有一只寿命小于180小时的概率?
?180?160?随机变量X:N(160,202),P(X?180)??????(1)?0.84134,
20??没有一只寿命小于180小时的概率为
P4(X?180)?(1??(1))4?(1?0.84134)4?0.00063368.
第九次作业
★1. 设离散型随机变量X具有概率分布律
X ?2 ?1 0 1 2 3 P 试求? E(X)? E(X2?5)? E(|X|)?
EX??xipi??2?0.1?1?0.2?1?0.3?2?0.1?3?0.1?0.4,
iEX2??xi2pi?(?2)2?0.1?(?1)2?0.2?12?0.3?22?0.1?32?0.1?2.2,
iE(X2?5)?EX2?5?7.2,
E|X|??|xi|pi?2?0.1?1?0.2?1?0.3?2?0.1?3?0.1?1.2.
i?0 x?0,2. 设随机变量X的概率密度为f(x)???x 0?x?1,求? (1)常数A? (2)X的数学期望?
?x??Ae x?1.1e?Ae?1,A?,
00122??1e??1e4(2) EX??xf(x)dx??x2dx??xe?xdx???2e?1?.
0021323★3. 设球的直径D在[a? b]上均匀分布?试求? (1)球的表面积的数学期望(表面积?D2)?
(1) 1????f(x)dx??xdx??1??Ae?xdx?(2)球的体积的数学期望(体积?D3)?
x2?dx?(a2?ab?b2); (1) E(?D)??ED???ab?a33bx???3??3dx?(a?b)(a2?b2). (2) E?D??ED???ab?a24?6?622b16★4. 设二维离散型随机变量(X? Y)的联合分布律为
X Y 1 2 3 4
求E(X)? E(Y)? E(XY)?
EX??xipig??2?(0.1?0.05?0.05?0.1)?2?(0.1?0.15?0.05?0.1)
i?2 0 2
0 ??2?0.3?2?0.35?0.1,
EY??yjpgj?1?(0.1?0.05?0.1)?2?(0.05?0.15)
j?3?(0.05?0.1?0.05)?4?(0.1?0.2?0.05)?2.65,
E(XY)??xi?yjpi,j
ij??2?(1?0.1?2?0.05?3?0.05?4?0.01)?2?(1?0.1?2?0.15?3?0.05?4?0.05) ??1.5?1.5?0.
?3e?3(y?1),y?1,?2x,0?x?1,fY(y)??★5. 设随机变量X和Y独立? 且具有概率密度为fX(x)??
y?1.?0,其它,?0,(1)求E(2X?5Y)? (2)求E(X2Y)?
2(1) EX??xfX(x)dx??2x2dx?,
00311EY????1??4yfY(y)dy??3ye?3(y?1)dy?,
1314或随机变量Z?Y?1:指数分布E(3),EZ?EY?1?,EY?,
3324E(2X?5Y)?2EX?5EY?2??5??8.
3311114222(2) EX??xfX(x)dx??2x3dx?,由X和Y独立得E(X2Y)?EX2EY???.
002233第十次作业
1. 设离散型随机变量X的分布列为
X P 试求? (1) D(X)? (2) D(?3X?2) ? ?2 0?1 ?1 0 1 2 3 (1) EX??xipi??2?0.1?1?0.2?1?0.3?2?0.1?3?0.1?0.4,
iEX2??xi2pi?(?2)2?0.1?(?1)2?0.2?12?0.3?22?0.1?32?0.1?2.2,
iDX?EX2?E2X?2.2?0.42?2.04. (2) D(?3X?2)?(?3)2DX?9?2.04?18.36.
?Ax2?2x,0?x?2,★2. 设随机变量X具有概率密度为f(x)??
?0,其他，试求? (1)常数A? (2)E(X)? (3) D(X)? (4) D(2X?3) ?
??289(1) 1??f(x)dx??(Ax2?2x)dx?A?4,解得A??.
??038??295(2) EX??xf(x)dx??x(?x2?2x)dx?.
??086944?5?19(3) EX2??x2f(x)dx??x2(?x2?2x)dx?,DX?EX2?E2X?????.
??0855?6?1801919(4) D(2X?3)?22DX?4??.
18045?2?x?y,0?x?1,0?y?1,★3. 设二维随机变量(X,Y)联合概率密度为f(x,y)??
?0,其他，试求? (1)X,Y的协方差和相关系数A? (2)D(2X?Y?1).
??13(1) fX(x)??f(x,y)dy??(2?x?y)dy??x,0?x?1,
??023由x,y的对称性fY(y)??y,0?y?1.
2??225?3?xfX(x)dx??x??x?dx??EY, ??0212????11?3?EX2??x2fX(x)dx??x2??x?dx??EY2,
??04?2?EX????11?5?11DX?EX?EX??????DY,
4?12?144222E(XY)??因此
?????????xyf(x,y)dydx??1xy(2?x?y)dydx?, 0?06111?5?1Cov(X,Y)?E(XY)?EXEY??????,
6?12?144Cov(X,Y)1?X,Y???.
11DXDY(2) 由随机变量和的方差公式D(X?Y)?DX?DX?2Cov(X,Y)得
D(2X?Y?1)?D(2X)?D(?Y)?2Cov(2X,?Y)59?22DX?(?1)2DY?2?2?(?1)?Cov(X,Y)?.
144★4. 设二维随机变量(X,Y)具有联合分布律
2Y X ?1 0 1 (1) X的分布列为
?2 0 ?1 0 0 1 2 0 试求EX,DX,EY,DY以及X和Y的相关系数? X pig i?1 0 1 由变量X分布对称得EX?0,或EX??xipig??1?0.45?0?0.45?1?0.45?0,
EX2??xi2pig?(?1)2?0.45?02?0.45?12?0.45?0.9,DX?EX2?E2X?0.9.
i(2) Y的分布列为
Y pgj (X,Y)取值关于原点中心对称
?2 ?1 0 1 2 由变量Y分布对称得EY?0,或EY??yjpgj??2?0.2?0.25?0.25?2?0.2?0,
i2222EY2??y2jpgj?(?2)?0.2?(?1)?0.25?1?0.25?2?0.2?2.1,
iDY?EY2?E2Y?2.1.
概率论与数理统计作业及解答
