expectation problem

[bravehera1]

10 balls are inserted (uniformly) into five jars ..each insertion is independent from the others.
compute E(X1*X2) where X1 and X2 describes the number of balls in each jar.

so far:
-P=Pr(Success from each jar poin of view=the ball was inserted into it)=1/5
-E(X1*X2)=E(E(X1*X2|X1))=E(X1*(10-X1)*P)=P(E(10X1-X1^2)=1OPE(X1)-E(X1^2)=10*(1/5)-10*9*((1/5)^2)+10*(1/5)) =4-90/25+2

what am i doing wrong ?

 

[Dason]

Do you mean X1 is the number placed in the first jar and X2 is the number placed in the second jar?

 

[bravehera1]

Do you mean X1 is the number placed in the first jar and X2 is the number placed in the second jar?

nop, x1 =number of balls inserted into the first jar..
in each step, a ball is inserted into one of the five jars.. so after 10 steps there are
x1 balls in jar 1,x2 balls in jar 2 ...

thanks for your reply.

 

[David1]

Define Xi- the number of balls that were inserted into jar i. (notice that Xi~Bin(10,1/5))
Define pi- the probability of a ball to be inserted into jar i. In this case, pi=1/5 for all i.

X=(X1,...,X5)
P=(p1,...,p5)

X~ Multinomial (10, P)
and Cov(Xi,Xj)=-n*pi*pj

http://en.wikipedia.org/wiki/Multinomial_distribution

use the following formula to find the expectation of the product:
E(X1*X2)-E(X1)*E(X2)=Cov(X1,X2)