4.1 Random Vartiables
- Example 1b. A life insurance agent has 2 elderly clients, each of whom has alife insuruance policy that pays \( \$ 100,000 \) upon death. Let \( Y \) be the cret that the vnumger one dies in the following year, and let \( O \) be the cerent that the ofler one dies in the following year. Assume that \( Y \) and \( O \) are independent, with respective probabilities \( P(Y)=.05 \) and \( P(O)=10 \). If X denotes the total amount of money (in units of \( \$ 100,000 \) ) that will be paid out this year to any of these clients' beneficiaries, then \( X \) is a random probaibitices
\[
\begin{array}{l}
P\{X=0\}=P\left(Y^{c} 0^{c}\right)=P\left(Y^{c}\right) P\left(O^{c}\right)=(.95)(.9)=.855 \\
P\{X=1\}=P\left(Y^{c}\right)+P\left(Y^{c} 0\right)=(.05)(.9)+(.95)(.1)=.140 \\
P\{X=2\}=P(Y 0)=(.05)(.1)=.005
\end{array}
\]
