# Thread: help needed computing variance

1. ## help needed computing variance

Suppose n balls are distributed at random into r boxes. Let Xi=1 if box i is empty and let Xi=0 otherwise.

a)Compute E(Xi) – answer: (1-1/r)^n
b)for i not equal j compute E(Xi,Xj) -answer: (1-2/r)^n
c) Let Sr denote the number of empty boxes. Write Sr=X1+X2...

and use result of a to compute E(Sr) -answer: r(1-1/r)^n
d) Use the result of a and b and compute Var(Sr)?????
I’m stuck on d… how do I compute var(Sr)???
thanks

2. Hello,
Originally Posted by adirh
Suppose n balls are distributed at random into r boxes. Let Xi=1 if box i is empty and let Xi=0 otherwise.

a)Compute E(Xi) – answer: (1-1/r)^n
b)for i not equal j compute E(Xi,Xj) -answer: (1-2/r)^n
c) Let Sr denote the number of empty boxes. Write Sr=X1+X2...

and use result of a to compute E(Sr) -answer: r(1-1/r)^n
d) Use the result of a and b and compute Var(Sr)?????
I’m stuck on d… how do I compute var(Sr)???
thanks
$\displaystyle \text{Var}(S_r)=\text{Var}(\sum_{i=1}^r X_i)=\sum_{i=1}^r \text{Var}(X_i)+2 \sum_{1\leqslant i<j\leqslant r} \text{Cov}(X_i,X_j)$
where $\displaystyle \text{Cov}(X_i,X_j)=\mathbb{E}(X_iX_j)-\mathbb{E}(X_i)\mathbb{E}(X_j)$