what is the meaning to evaluate the variance of probability destribution for insurance in general?
What does it do(setting the price, estimate cash reserve or else), also does evaluate the variance of probability destribution directly help to setting the pricing or estimate the cash reserve?
Imagine selling (i) $10^5$ dollar life insurance policies to $10^4$ people, at a price that makes expected profit on each policy $\$5000$; or (ii) selling a $10^9$ dollar life insurance policy to $1$ person, at a price that brings expected profit $5×10^7$.
The expected profits on (i) and (ii) are the same. But the variance in the profit on transaction (ii) is much higher. The probability the insurance company gets ruined is much higher in (ii). With less dramatic numbers, substantially higher cash reserves will be needed if one is selling policies of the general type (ii) than if one sells policies of type (i). Higher variance is associated with the need for greater access to cash reserves.
There are subtler versions, such as making variance smaller in flood insurance by selling over a wide geographical area. In investing, diversification of portfolio can be thought of as a variance lowering device.