Question

The number of goals Alice scores in a soccer game is a Poisson(2.2) random variable. Her coach awards her 2k candies if she scores k goals in a game. For example, if she scores 1 goal she gets 2 candies, if she scores 2 goals she gets 4 candies, if she scores 3 goals she gets 8 candies, and so on. If Alice does not score any goals, she does not get any candies. What is the expected number of candies Alice gets in a game?

Answer 1

P(k goals in match) = (2.2^k*exp[-2.2])/k!

expected number of candies Alice gets in a game = $\displaystyle\sum_{k=1}^\infty$ [(2.2^k*e^-2.2)/k!*2^k]=

$\displaystyle\sum_{k=1}^\infty$ [(2.2^k*e^-2.2)/k!*2^k] = e^-2.2 $\displaystyle\sum_{k=1}^\infty$ [(4.4^k)/k!] = e^-2.2 ( $\displaystyle\sum_{k=0}^\infty$ [(4.4^k)/k!] - 4.4⁰/0!) = e^-2.2 (e^4.4 - 1) = 8.91