Question

设袋中有2只红球和3只白球,现n个人轮流摸球,每个摸出2只球后将球放回袋中再由下一人摸,求n个人总共摸到红球数的数学期望

Answer 1

设X,Y分别是每个人摸到红球数和所有人共摸到红球数,Y=∑X,其中X1,...,Xn相互独立.

对于每个人而言,P(X=0)=3/5 * 2/4=3/10

P(X=0)=3/5 * 2/4 + 2/5 * 3/4=6/10

P(X=2)=2/5 * 1/4=1/10

E(X) = 0*0.3 + 1*0.6 + 2*0.1 = 0.8

E(X^2) = 0*0.3 + 1*0.6 + 4*0.1 = 1

D(X) = 1 - 0.8*0.8 =0.36

所以,E(Y=∑X) = E(Y=X1+X2+...+Xn) = E(X1)+E(X2)+...+E(Xn) = nE(X) = 0.8n

D(Y) = nD(X) = 0.36n