Quantitative Finance

Probability Theory

Continuous Random Variables - Problem Solving

         

Suppose there are two new effective regimens (regimen AA and regimen BB) 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 AA results in a better survival rate. How much longer (in months) is the life expectancy when using regimen AA than that when administrating regimen BB?

Details and Assumptions

  • A survival curve plots the fraction of survived patients according to elapsed time.
  • a=1324a=\frac{1}{324} and b=123.b=\frac{1}{23}.
  • The figure is not drawn to scale.

Suppose there is a bus stop where the bus stops every 77 minutes. If Tom goes to that bus stop at an arbitrary time, what is the variance of the time Tom has to wait for the bus?

Let continuous random variables XX and YY have the joint probability density function f(x,y)=19xyf(x,y)=\frac{1}{9}xy for 0x20\leq x\leq2 and 0y3.0\leq y\leq3. Find the conditional probability P(X1Y2).P(X\leq1|Y\leq2).

If the probability distributions of continuous random variables X X and Y Y are defined in the domains 0x1 0 \le x \le 1 and 0y10\le y\le1 as follows: fX(x)=6x5,fY(y)=10y9, f_X(x) = 6x^{5} , f_Y(y) = 10 y^{9}, what is the covariance between X X and Y? Y ?

Suppose that the life expectancy XX of a cat species has the probability density function f(x)=kx(x20)f(x)=kx(x-20) for 0X20.0\leq X\leq20. John the farmer decided to breed this species, and bought hundreds of these newborn cats. If John currently has 20002000 cats, all of which are 1010 years old, how many of these cats will die before becoming 1414 years old?

Note: Ignore reproduction of the cats.

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