PROBLEM 1. (Run, Baby, Run, 10 points) To support the many runners that participated in the
Boston marathon, you decided to set up a water stand to pass out cups of water to thirsty runners.
Throughout the day, you noticed that a total of 300 runners, 500 joggers, and 400 walkers passed
by your stand. Each type of marathoner has a different probability of needing water, independent of
other marathoners, as follows:
• Runners want two cups of water with probability 0.6, and zero with the remaining probability.
• Joggers want one cup of water with probability 0.7, and zero with the remaining probability.
• Walkers want one cup of water with probability 0.2, and zero with the remaining probability.
Let the random variables $R$, $J$, and $W$ be the number of runners, joggers, and walkers wanting
water, and $C$ be the total number of cups they want.
(a) Identify the distributions for $R$, $J$, and $W$, and write an equation for $C$ in terms of these variables.
(b) Compute $E(C)$ and $Var(C)$.
Suppose you have enough water for 1000 cups at the start of the marathon.
(c) Using Markov's Inequality, give a lower bound to the probability that you will have enough water
for all the marathoners that pass your stand.
(d) Use Chebyshev's Inequality to give a tighter bound on the same event as (c).
