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A mortgage company manages 1, 000 mortgages. Mortgages default at a rate of 10 per year...

A mortgage company manages 1, 000 mortgages. Mortgages default at a rate of 10 per year (you can assume this rate remains constant and that defaults can be described as a Poisson process). Every defaulted mortgage gives a loss of $100, 000. Every mortgage that doesn’t default gives a yearly profit of $1, 000. Let X be the number of mortgages that default in a year and let F be the total earnings in a year.

1) Find the yearly earnings F as a function of X.

2) Find the expected yearly earnings, E[F].

3) Find the probability that earnings are negative, P(F < 0).

4) Find the probability that there are exactly five defaults in a given year, given that there have been no defaults in the first half of the year.

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