Quantitative Finance
# Probability Theory

Suppose there are two new effective regimens (regimen \(A\) and regimen \(B\)) that can be used for treating advanced pancreatic cancer. Dr. Johnson wants to decide which regimen to use for his patient. According to the national data of pancreatic cancer patients, the survival curves when using each regimen are as shown in the figure above. This data suggests that regimen \(A\) results in a better survival rate. How much longer (in months) is the life expectancy when using regimen \(A\) than that when administrating regimen \(B\)?

**Details and Assumptions**

- A survival curve plots the fraction of survived patients according to elapsed time.
- \(a=\frac{1}{324}\) and \(b=\frac{1}{23}.\)
- The figure is not drawn to scale.

Suppose that the life expectancy \(X\) of a cat species has the probability density function \[f(x)=kx(x-20)\] for \(0\leq X\leq20.\) John the farmer decided to breed this species, and bought hundreds of these newborn cats. If John currently has \(2000\) cats, all of which are \(10\) years old, how many of these cats will die before becoming \(14\) years old?

**Note:** Ignore reproduction of the cats.

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