Question

概率论 期望对一台仪器进行重复测试,直到发生故障为止,假定测试是独立的,每次测试发生故障的概率为0.1,求次数X的数学期

Answer 1

1 2 ...k .

0.1 0.9*0.1 ...0.9^(k-1)*0.1 .

E(X) =1*0.1 + 2*0.9*0.1 +...+ k*0.9^(k-1)*0.1+...

E(X) =0.1(1 + 2*0.9 +...+ k*0.9^(k-1)+...)

10E(X)=1 + 2*0.9 +...+ k*0.9^(k-1)+...A

0.9*10E(X)=0.9 + 2*0.9^2 +...+(k-1)*0.9^(k-1)+ k*0.9^k+...

9E(X)=0.9 + 2*0.9^2 +...+(k-1)*0.9^(k-1)+ k*0.9^k+...B

A-B

E(X)=1+ 2*0.9 -0.9 +3*0.9^2-2*0.9^2 +...k*0.9^(k-1)-(k-1)*0.9^(k-1)+...

E(X)=1+0.9+0.9^2+...+0.9^(k-1)+..

E(X)=1/(1-0.9)=10