Problem

An insurance company offers its policyholders a number of different premium payment opti...

An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X = the number of months between successive payments. The cdf of X is as follows:

a. What is the pmf of X?

b. Using just the cdf, compute P(3 ≤ X ≤ 6) and P( 4 ≤ X).

Step-by-Step Solution

Solution 1

Step #1 of 6

(a)

Given below is the cumulative probability distribution of number of months between successive payments.

The cumulative distribution function is

$$ \begin{aligned} F(x) &=0 & & x<1 \\ &=0.30 & & 1 \leq x<3 \\ &=0.40 & & 3 \leq x<4 \\ &=0.45 & & 4 \leq x<6 \\ &=0.60 & & 6 \leq x<12 \\ &=1 & & 12 \leq x \end{aligned} $$

Step #2 of 6

Probability mass function of $x$ is (using the cumulative distribution function)

$$ \begin{aligned} p(0) &=0 \quad(\text { since } F(x)=0 \quad x<1) \\ p(1) &=F(1)-F(0) & \\ &=0.3-0 \quad &(F(x)=0.30 \text { if } 1 \leq x<3) \\ &=0.3 \end{aligned} $$

using the $\operatorname{CDF}(F(x)=0.30$ if $1 \leq x<3)$

$$ p(2)=0 $$

$$ \begin{aligned} p(3) &=F(3)-F(2) \\ &=0.4-0.3 \\ &=0.1 \end{aligned} $$

Step #3 of 6

$\begin{aligned} p(4) &=F(4)-F(3) \\ &=0.45-0.4 \\ &=0.05 \\ & \text { using the } \operatorname{CDF}(F(x)=0.45 \text { if } 4 \leq x<6) \\ p(5) &=0 \\ p(6) &=F(6)-F(5) \\ &=0.60-0.45 \\ &=0.15 \\ & \text { using the } \operatorname{CDF}(F(x)=0.6 \text { if } 6 \leq x<12) \\ p(7) &=0 ; p(8)=0 ; p(9)=0 ; p(10)=0 ; p(11)=0 \end{aligned}$

Step #4 of 6