2 5 随机变量函数的分布

1、2.5 随量函数的分布一、离散型随量函数的分布-1X012例 已 知r.v.X 的分布是p0.20.10.30.4求Y = 4X + 1,的分布.Z = X 2解- 1X012-31591Z = X 2104P0.20.10.30.444Y = 4X + 1Z014P0.10.50.Y- 3159P0.20.10.30.2,例 已知 r.v.X 的分布是 P X = n =n = 1, 2, 3,. p3nY= sin 2 X , 求Y 的概率分布.解YPY= P X = 2 + P X = 4 + P X = 6 + . + P X = 2n + . 2 322222= 1=+ . + .=

2、32343632n1 - 149X12345678.n.= sin p X 210-1010-10sin n p2P2222332333422223536373823nYp-101= 0 = Psin 2 X = 0P14解YX = 1p2PY = 1 =Psin= P X = 1 +P X = 5 + P X = 9 + . +P X = 4n + 1 + .22222= 273=+ . + . =34n-3414 3 402740PY = -1 =1 -=Y-101P 312740440X12345678.n.= sin p X 210-1010-10sin n

3、p2P2222332333422223536373823n二、 连续型随量 函数的分布( y5 )21例 设随量f( x) =- x +,Xp(1 + x2 )XY = 2X + 5, 求Y 的密度函数.( y) = PY y= P 2 X + 5y= PX y - 5 解= FY2 y-5 y-5 2-1f X ( x)dx = dx2- p (1 + x)2 1dx =1f( y) =F ( y) = 12 y-52()YY22 - p (1 + x) 2y-5p1 +2=p4 + ( y - 5)22( y ) =r.v.Y - y ,fp 4 + ( y - 5 )2 Y5 z )1例

4、 设随量f( x) =- x +X,p(1 + x2 )X求 Z 的密度函数.Z = 5 - X ,FZ (z) = PZ z= P 5 - X解 z= P X 5 - z15 - z f5 - z= 1 -P X 5 - z = 1 - - p (1 + x) -( x)dx = 1 -dx2X5 - z 1dx f(z) = F (z) = 1 - -p (1 + x2 )ZZdx115 - z( (-1)= -p 1 +(5 - z ) -p (1 + x22)1(z ) =r.v.Z - z 0 时，FY ( y) =P Y y = P aX + b y( x-m)2= P X = y

5、 - b-y - b12s2a edxa2ps-()2y - b-m( x-m)2a-12psa- y - b12s2f( y) =F( y)2s2dx =eeaYY2ps- y - (am + b)2-= 1eY N (am + b,a2s 2)2( sa )22p(sa)s a定理2.6设随量 X N (m, s 2 )Y= aX + b其中 a, b 为常数，且 a 0,则 Y N (am + b, a2s 2 )证 当a 0 时，FY ( y) = P Y y= P aX + by= P aX y - b= P Xy - by - b= 1 - PX aa( x-m)2Y N (am

6、+ b,a2s 2 ) y - b-12= 1 -a2ps e2sdx-( x-m)2 y - b-1f( y) = F ( y)2s21 -a 2ps edxYY-()2 y - ba y -(am + b)2-m-112( sa )2e=2s2= -a e2p2ps(s a)2(ln y )01例 设随量f( x) =- x 0解 FY ( y) = PY y= P e X y = P X ln y,y 0 时，1p (1 + x2 )ln y f( y) = P X ln y=ln y - -( x)dx =FdxYXy 0时，= 11f( y) =F ( y)= ln y 1dxYYp

7、 1 +( ln y )2p (1 +2-yx)y 0f( y) = 0Y = e X 1Y py 1 + (ln y)2 1例 设随量f( x) =- x +X,p(1 + x2 )XZ = X 2 , 求Z的密度函数.(z) = PZ z= P X 2 z解FZP (F )= 0z 00 z dx解 F(z) = PZ z = P X 2z=P XZz 0 时， z1p (1 + x2 )-1p (1 + x2 )0z0F(z)= -dx +Zz 0 时，z 1dx f(z) = F (z)= -z 1dx +0p (1 + x2 )ZZp (1 + x2 )0z )+ 1(z ) = 1

8、 2= - 1(-p(1 + z )p(1 +z )p(1 + z ) 2zz 0 为Z = X 2 的密度函数. 01fZ (z) = 1,p(1 + z )z(e y )4 x + 1 ,0 x 1其它例 设 r.v.X f X ( x) = 30,Y = ln X ,求Y的密度函数.解 FY ( y) = PY y= P ln X y = P0 0031y =fX ( x)4 e y + 1e y ,13y 0f( x) =21ex + e- x例 设- x +,r.v.XpX-1,若x 0Y= g ( X ), 其中 g( x) = 求Y的分布. 1若x 0- 1121 若X0若X00-1,Y 1 解Y = g ( X ) = 22 1,10P Y = -1= P X 0 =( x)dx =fdxXex + e- xp-= 2 0 1= 2= 2ex0-0xxdx+ 1-1 + (ex )2 d earctanepppe2 x-2(arctan1- arctan 0)= 1连续型随量的函数=可能是离散型随量.p20 x 1其它f( x) = 1,Z =ln X例设r.v.X X求 Z的密度函数.0,解z 0 时,0 e- z 1, 1,ezezez1f X ( x)dx = FZ (z) =P e- z X ez=+e- z1dx0dx