Probability mass function

Question

An insurance agent has two clients each of whom has a life insurance policy that pays 100000 dollars upon death. Their probabilities of dying this year are .05 and .1. Let X denote the total amount of money that will be paid this year to the clients beneficiaries. Assuming that the event that client 1 dies is independent of the event that client 2 dies, determine the probability distribution (probability mass function ) of X also EX.

There is no cost given for the policy. I know dying gets prob .05 and .1, then living gets prob .95 and .9. EX would be 100000*.05 - cost*.95 etc. But not sure Im thinking about this correctly.

Answer 1

You don't need to know the cost of the policy, because the only outcome of interest is the random amount of the death benefit paid this year.

To this end, it is instructive to explicitly write out the probability distribution of $X$:

$$\begin{align*} \Pr[X = 0] &= (1 - 0.05)(1 - 0.1) \\ \Pr[X = 10^5] &= (0.05)(1 - 0.1) + (1 - 0.05)(0.1) \\ \Pr[X = 2 \times 10^5] &= (0.05)(0.1). \end{align*}$$

This is because in order to observe $X = 0$, neither policyholder may die; if $X = 10^5$, exactly one policyholder dies (i.e., the first survives and the second does not; or the first dies and the second does not, and these are mutually exclusive events so the probability of either such outcome occurring is the sum of the two individual probabilities). And finally, if $X = 2 \times 10^5$, then both policholders die.

Then the expectation is simply $$\operatorname{E}[X] = 0(0.95)(0.9) + 10^5 ((0.05)(0.9) + (0.95)(0.1)) + 2(10^5)(0.05)(0.1).$$

Of course, this could also be computed by exploiting the independence of events of the two policyholders' life statuses. The expected payout on the first policyholder is simply $10^5 (0.05)$, and the second is $10^5 (0.1)$, so the expected total payout is simply the sum of the individual expected payouts due to the linearity of expectation.