Solution: 2015 Winter Final - 22

Author: Michiel Smid

Question

One of Lindsay and Simon is chosen uniformly at random. The person that is chosen wins \$100. Define the random variables $L$ and $S$ as follows:

and

Which of the following is true?

(a)

$\mathbb{E}(L) = 100$ and $\mathbb{E}(\max(L,S)) = 50$.

(b)

$\mathbb{E}(L) = 100$ and $\mathbb{E}(\max(L,S)) = 100$.

(c)

$\mathbb{E}(L) = 50$ and $\mathbb{E}(\max(L,S)) = 50$.

(d)

$\mathbb{E}(L) = 50$ and $\mathbb{E}(\max(L,S)) = 100$.

Let’s have a look, I guess

To determine $ \mathbb{E}(L) $, we need to determine the probability that Lindsay wins
$ Pr(L=100) = \frac{1}{2} $
$ Pr(L=0) = \frac{1}{2} $
$ \mathbb{E}(L) = 100 \cdot \frac{1}{2} + 0 \cdot \frac{1}{2} $
$ \mathbb{E}(L) = 50 $
To determine $ \mathbb{E}(S) $, we need to determine the probability that Simon wins
$ Pr(S=100) = \frac{1}{2} $
$ Pr(S=0) = \frac{1}{2} $
$ \mathbb{E}(S) = 100 \cdot \frac{1}{2} + 0 \cdot \frac{1}{2} $
$ \mathbb{E}(S) = 50 $
Let's write out the set for $ \text{max}(L,S) $
$ \text{max}(L,S) = { (100, 0), (0, 100) } $
$ \mathbb{\text{max}(L,S)} = 100 \cdot \frac{1}{2} + 100 \cdot \frac{1}{2} $
$ \mathbb{\text{max}(L,S)} = 100 $