Expect value question about a patient's hospitalisation

Question

I'm trying to solve this question

The answer is $\$360$, I tried to use the expected value definition, $E(X)=\Sigma x f(x)$, but I don't know where I put the $\$200$ and $\$100$ numbers.

Answer 1

Define a new random variable $P$ to be the payout. It will have probability mass function:

$$g(p)=\begin{cases}f(1), & p=200 \\ f(2), & p=400 \\ f(3), & p=500 \\ f(4), & p=600\end{cases}$$

Now, you are looking for $E[P] = \sum pg(p)$.

Answer 2

You will pay \$400 for a two day stay, \$500 for three days and \$600 for a four day stay. $$E(p)=200\frac{4}{10}+400\frac{3}{10}+500\frac{2}{10}+600\frac{1}{10}$$