Question

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

\( F(x)=\begin{cases} 0,\hspace{7mm} x < 1 & \quad \\ 0.10, \hspace{2mm}1\leq x < 3 & \quad \\ 0.40, \hspace{2mm}3 \leq x < 7 & \quad \\ 0.80, \hspace{2mm} 7 \leq x < 12 & \quad \\ 1,\hspace{7mm} 12 \leq x: & \quad \end{cases} \)

(a) What is the pmf of X?

(b) Using just the cdf, compute \( P(3 \leq X \leq 6)\hspace{2mm} \) and\( \hspace{2mm} P(X \leq 6): \)

 

Answer 1

Solution

(a) What is the pmf of X

Probability Mass Function (pmf) is :

(b) Using just the cdf, compute \( P(3 \leq X \leq 6)\hspace{2mm} \) and \( P(6 \leq X) \)

\( \implies P(3 \leq X \leq 6)=P(X \leq 6)-P(X<3) \)

                                    \( =P(6 \leq X)-P(X\leq 1) \)

                                    \( =F(3)-F(1)=0.3 \)

\( \implies P(6\leq X)=1-P(X<6)=1-P(X\leq 3)=1-F(3)=0.6 \)

Therefore.

a).

b).\( P(3 \leq X \leq 6)=0.3\hspace{2mm},P(6\leq X)=0.6 \)