Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box i with probability

p_(i), sum_(i=1)^(5)p_(i)=1

Find the expected number of boxes that have exactly 1 ball.

p_(i), sum_(i=1)^(5)p_(i)=1

Find the expected number of boxes that have exactly 1 ball.

1 Answers

Best Answer

Define indicator random variables J_(i), i=1,...,5 that marks whether i^(th) box has one and only one ball or not.
Observe that
P(J_(i)=1)= (begin(array)/(c)10 1 end(array) )p_(i)(1-p_(i))^(9)
since we have that i^(th) box has one and only if we choose one ball out of ten and put it into that box, and all other balls we put in some of remaining boxes. Hence, the number of empty boxes can be expressed as M= sum_(i=1)^(5)J_(i), so, using the linearity of expectation, we get
E(M)= sum_(i=1)^(5)E(J_(i))= sum_(i=1)^(5)P(J_(i)=1)= sum_(i=1)^(5)10*p_(i)(1-p_(i))^(9)