Moments of a binomial loss distribution

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Suppose a portfolio has  equally-sized exposures. Each one is independent and has a probability  of creating a unit loss (and a probability  of creating a zero loss), with  the same for each exposure, meaning that the portfolio loss, , is distributed according to a binomial distribution, i.e.:



The mean and the variance of the portfolio loss distribution can be found as follows. We note that:



The mean of the loss distribution is given by:








The variance of the loss distribution is:



Thus binomial distribution has mean  and variance .


As , the Central Limit Theorem CLT implies that the binomial distribution tends to a normal distribution with the same mean and variance, i.e. to  where  is the cumulative normal distribution.


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