Question

The Occupancy Problem. You have $n$ bins and $m$ balls. For each ball, you will throw it into one of the bins uniformly at random. Thus, if we define $A_{ij}$ to be the event that ball i lands into bin $j$, then: $Pr[A_{ij}] = 1/n$ Let $X_i$ be a random variable representing the number of balls in bin $i$ for any $i in {1, ..., n}$. (a) Show that $E[X_i] = m/n$. (b) For any fixed $t$, express $Pr[X_i ge t]$ as a binomial probability distribution in terms of $m$. (c) Use Chernoff's bound to find $Pr[X_i ge t]$ (in terms of $t$ only) when $n = m$. What value of $t$ would make the probability at most $n^{-c}$ for some constant $c > 2$. (d) Use the Union bound to find the probability that at least one of the $n$ bins has more than $t$ balls, where $t$ is the value you computed in part (c).

          The Occupancy Problem. You have $n$ bins and $m$ balls. For each ball, you will throw
it into one of the bins uniformly at random. Thus, if we define $A_{ij}$ to be the event that ball
i lands into bin $j$, then:
$Pr[A_{ij}] = 1/n$
Let $X_i$ be a random variable representing the number of balls in bin $i$ for any $i in {1, ..., n}$.
(a) Show that $E[X_i] = m/n$.
(b) For any fixed $t$, express $Pr[X_i ge t]$ as a binomial probability distribution in
terms of $m$.
(c) Use Chernoff's bound to find $Pr[X_i ge t]$ (in terms of $t$ only) when $n = m$.
What value of $t$ would make the probability at most $n^{-c}$ for some constant $c > 2$.
(d) Use the Union bound to find the probability that at least one of the $n$ bins
has more than $t$ balls, where $t$ is the value you computed in part (c).

The Occupancy Problem. You have n bins and m balls. For each ball, you will throw
it into one of the bins uniformly at random. Thus, if we define Aij to be the event that ball
i lands into bin j, then:
Pr[Aij] = 1/n
Let Xi be a random variable representing the number of balls in bin i for any i in 1, ..., n.
(a) Show that E[Xi] = m/n.
(b) For any fixed t, express Pr[Xi ge t] as a binomial probability distribution in
terms of m.
(c) Use Chernoff's bound to find Pr[Xi ge t] (in terms of t only) when n = m.
What value of t would make the probability at most n^-c for some constant c > 2.
(d) Use the Union bound to find the probability that at least one of the n bins
has more than t balls, where t is the value you computed in part (c).

Added by Brenda H.

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