# Central limit theorem problem

1. Dec 14, 2006

### twoflower

Hi all, I just wrote a test from probability and had troubles doing this problem:

1. The problem statement, all variables and given/known data

The assurance company makes an insurance for 1000 people of the same age. The probability of death during the year is 0.01 for each of them. Each insured person pays 1.200 dollars a year. In case of death the assurance company pays 80.000 dollars. Figure out:

a) expected income of the assurance company

b) approximate probability of that the assurance company will suffer a loss, knowing that $u_{0.9} = 1.28$, $u_{0.95} = 1.64$ and $u_{0.975} = 1.96$, where $u_{\alpha}$ is $\alpha$-quantil of the normal distribution $N(0,1)$.

The first one was quite easy, it's just the expected value of binomial distribution.

But I couldn't solve the second one. I know it's an example of using the Central limit theorem, but I was confused of what to actually put in the theorem...I computed the 'critical value' A in the sense that if more than A people die during the year, the assurance company will suffer a loss. So I got

$$P(\mbox{assurance companny will suffer a loss}) = P(\mbox{number of people who died} > A) = 1 - P(\mbox{number of people who died} < A)$$

$$= 1 - P\left(\sum_{i=1}^{1000} X_{i} < A\right)$$

where $\sum_{i=1}^{1000} X_{i} \sim Bi(1000, 0.01)$

Then I normalized the probability so that I got $N(0,1)$. Is it the right approach? Because I got strange numbers which didn't seem anyhow related to those we had been given (those quantils).

Thank you for any help.