1. **Problem statement:** Given a discrete random variable $X$ representing the number of days a patient stays in the hospital with probability mass function (pmf) $f(x) = \frac{5 - x}{100}$ for $x = 1,2,3,4$, and payment scheme: 200 for each of the first two days and 100 for each additional day, find the expected total payment. 2. **Formula and rules:** The expected value $E(Y)$ of a function $Y = g(X)$ is $E(Y) = \sum_x g(x) P(X=x)$ for discrete $X$. 3. **Calculate payments $Y$ for each $x$:** - For $x=1$: $Y = 200 \times 1 = 200$ - For $x=2$: $Y = 200 \times 2 = 400$ - For $x=3$: $Y = 200 \times 2 + 100 \times (3-2) = 400 + 100 = 500$ - For $x=4$: $Y = 200 \times 2 + 100 \times (4-2) = 400 + 200 = 600$ 4. **Calculate probabilities:** - $P(X=1) = \frac{5-1}{100} = \frac{4}{100} = 0.04$ - $P(X=2) = \frac{5-2}{100} = \frac{3}{100} = 0.03$ - $P(X=3) = \frac{5-3}{100} = \frac{2}{100} = 0.02$ - $P(X=4) = \frac{5-4}{100} = \frac{1}{100} = 0.01$ 5. **Calculate expected payment:** $$ E(Y) = \sum_x Y(x) P(X=x) = 200 \times 0.04 + 400 \times 0.03 + 500 \times 0.02 + 600 \times 0.01 $$ $$ = 8 + 12 + 10 + 6 = 36 $$ **Final answer:** The expected total payment is 36.