独立三次实验,第i次实验成功的概率pi=i/(i+1),i=1,2,3.求三次实验中成功的次数X的概率分布.(高数题)

题目

独立三次实验,第i次实验成功的概率pi=i/(i+1),i=1,2,3.求三次实验中成功的次数X的概率分布.(高数题)
(0 1/24 1 1/4 2 11/24 3 1/4)

答案

由已知p1=1/2,p2=2/3,p3=3/4
则x=0时,三次都没成功,P(x=0)=(1-1/2)*(1-2/3)*(1-3/4)=1/24
x=1时,三次中有1次成功.
P(x=1)=(1/2)*(1-2/3)*(1-3/4)+(1-1/2)*(2/3)*(1-3/4)+(1-1/2)*(1-2/3)*(3/4)
=(1/2)*(1/3)*(1/4)+(1/2)*(2/3)*(1/4)+(1/2)*(1/3)*(3/4)
=6/24
=1/6
x=3时,三次都成功.
P(x=3)=(1/2)*(2/3)*(3/4)=6/24=1/4
∴ P(x=2)=1-P(x=0)-P(x=1)-P(x=3)=1-1/24-1/4-1/4=11/24
然后就得到你所给的答案.