Re: [問題] 2題統計機率問題求解

[BF3716 (JS)]

1.設隨機變數x，有下列之機率分配，則E(X^2+2X+1)=?
X     0    1   2   3
f(x)  4P   6P  6P  4P
已知V(X)=2 V(Y)=3且X Y為獨立的隨機變數，則同上題，Ｖ(3X-2Y)=?

[a].  機率和為1 (4P+6P+6P+4P =1 )，先解出P值 = 0.05

      X     0   1   2   3
     f(x)  0.2 0.3 0.3 0.2

E(X) = 0*0.2 +1*0.3 +2*0.3 +3*0.2 = 1.5
E(X^2)= (0^2)*0.2 +(1^2)*0.3 +(2^2)*0.3 +(3^2)*0.2 = 3.3
E(X^2+2X+1)=E(X^2)+2E(X)+1= 3.3 +2*1.5 +1 = 7.3

[b]. V(3X-2Y)= 9V(X)-12COV(XY)+4V(Y)
             = 9V(X)+4V(Y)
             = 9*2+4*3
             = 30                      #(同上題?? 由上述條件Var(x)不等於2)


2.袋中有10個號碼球，１號球４個，２號球３個，３號球２個，４號球１個
　從袋中隨機抽取一球，令Ｘ表抽到的號碼球數，則Ｐ(X<=2)=?
 則期望值，變異數分別為多少？

      X     1   2   3   4
     f(x)  0.4 0.3 0.2 0.1

[a]. Ｐ(X<=2)= P(X=1)+P(X=2)= 0.4+0.3= 0.7

[b]. E(X) = 1*0.4 +2*0.3 +3*0.2 +4*0.1 = 2

[c]. E(X^2)= (1^2)*0.4 +(2^2)*0.3 +(3^2)*0.2 +(4^2)*0.1 = 5

     Var(X)= E(X^2)-E(X)^2 = 5-2^2 = 1


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