Solution: 2014 Fall Final - 24
Author: Michiel SmidQuestion
We flip a fair coin independently $n$ times. Define the random variable
- X = twice the number of heads minus three times the number of tails.
(a)
$-n$
(b)
$n$
(c)
$-n/2$
(d)
$n/2$
Solution
- Let A be the event that we flip a head
$ Pr(A) = \frac{1}{2} $ - Let B be the event that we flip a tail
$ Pr(B) = \frac{1}{2} $
$ \mathbb{E}(X) = 2 \cdot \mathbb{E}(\text{number of heads}) - 3 \cdot \mathbb{E}(\text{number of tails}) $
$ \mathbb{E}(X) = 2 \cdot \sum_{i=0}^{n} Pr(A) - 3 \cdot \sum_{i=0}^{n} Pr(B) $
$ \mathbb{E}(X) = 2 \cdot n \cdot \frac{1}{2} - 3 \cdot n \cdot \frac{1}{2} $
$ \mathbb{E}(X) = n - \frac{3}{2}n $
$ \mathbb{E}(X) = \frac{2n}{2} - \frac{3n}{2} $
$ \mathbb{E}(X) = \frac{-n}{2} $