1. C ONDITIONAL P ROBABILITY AND C ONDITIONAL E XPECTATION Name/ Taha Ben Omar ID/ 145074

2. As noted previously, conditional expectations given that Y = y are exactly the same as ordinary expectations except that all probabilities are computed conditional on the event that Y = y. As such, conditional expectations satisfy all the properties of ordinary expectations. For instance, the analog of

3. Is that If E[X|Y,W] is defined to be that function of Y and W that, when Y = y and W = w, is equal to E[X|Y = y,W = w], then the preceding can be written as E[X|Y] = E"E[X|Y,W]!! Y#

4. Example 3.29 An automobile insurance company classifies each of its policyholders as being of one of the types i = 1,..., k. It supposes that the numbers of accidents that a type i policyholder has in successive years are independent Poisson random variables with mean λi, i = 1,..., k. The probability that a newly insured policyholder is type i is pi, k i=1 pi = 1. Given that a policyholder had n accidents in her first year, what is the expected number that she has in her second year? What is the conditional probability that she has m accidents in her second year?

5. Solution: Let Ni denote the number of accidents the policyholder has in year i, i = 1, 2. To obtain E[N2|N1 = n], condition on her risk type T.

6. where the final equality used that The conditional probability that the policyholder has m accidents in year 2 given that she had n in year 1 can also be obtained by conditioning on her type.

7. Another way to calculate P{N2 = m|N1 = n} is first to write and then determine both the numerator and denominator by conditioning on T. This yields

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