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Thread: help needed computing variance

  1. February 27th 2009, 07:24 AM #1
    adirh
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    help needed computing variance

    Please help me solve this..
    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
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  2. February 27th 2009, 09:44 AM #2
    Moo
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    Hello,
    Quote Originally Posted by adirh View Post
    Please help me solve this..
    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
    \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 \text{Cov}(X_i,X_j)=\mathbb{E}(X_iX_j)-\mathbb{E}(X_i)\mathbb{E}(X_j)
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