Question

Answer number 7 Below

7. Suppose Ω = {1, 2, 3, . . .} and P({n})-n(n+1) for n € Ω. What should c be (or is it impossible to determine such a c)? 8. This problem is similar to the Suppose you experience a loss of L dollars, and the loss is covered by insurance with a deductible of $1,000. (This problem is similar to the problem on the first homework with 7 items in stock and a demand D.) a) How much money would u receive from the insurance company if

Answer 1

Given the sample space as $\Omega =\left \{ 1,2,3,... \right \}$ . And the discrete probabilities as $P\left ( \left \{ n \right \} \right )=\frac{c}{n\left ( n+1 \right )},n\in \Omega$ .

The condition for pdf is

$\sum_{n=1}^{\infty }P\left ( \left \{ n \right \} \right )=1\\ \sum_{n=1}^{\infty }\frac{c}{n\left ( n+1 \right )}=1\\ c\sum_{n=1}^{\infty }\left (\frac{1}{n}-\frac{1}{n+1} \right )=1\\ c\left ( 1-\frac{1}{2} +\frac{1}{2} -\frac{1}{3}+\frac{1}{3}-\frac{1}{4} +... \right )=1\\ {\color{Blue} c=1}$

Note that $\sum_{n=1}^{\infty }\left (\frac{1}{n}-\frac{1}{n+1} \right )$ is a telescoping series.