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Problem

1. Consider what happens when you throw two dice. You can ask the question "what's the probability of getting a total of 8 when the first one is 5?. Or in more generality: what's the probability of getting a total of $T$ when the first one is $m$?. The answer is a function of two variables, $T$ and $m$. You could call the result $P(m,T)$. How do you define averages in this case? For example, what's the average value of $m T$.

You could determine $\langle m T \rangle$ by throwing the dice repeatedly taking the number of the first die and multiplying by the total. What you get should be equivalent to

\begin{displaymath} \langle m T \rangle = \sum_m \sum_T m T P(m,T) \end{displaymath} (1.36)

but that's a sum over a lot of possibilities, all possible values of $m$ and $T$. Instead use independence to solve this problem in analogy to what we just did for the two dice.

next up previous contents
Next: Average of a Binomial Up: Averages in involving multiple Previous: Averages for independent variables   Contents

josh 2010-10-20