MODELING EXAMPLE $(1)$

$-$ Before introducing a new car insurance policy, an insurer wants to study consumer behaviour relative to the policies already available in the market. Suppose there are $3$ main players in the insurance industry, say player A$,$ B and C$.$ The consumers either buy their policy from the same provider for a few years, or change their brands every now and then when a superior brand is introduced. After conducting a survey for a few years, the consumer brand$-$switching behaviour can be summarized as in the above table.

$-$ What is the expected number of years consumer would remain loyal to a specific firm?

$-$ After a certain number of years, what are the mean and variance of the consumers buying policies from certain player?