In the Bernoulli mixture model, we randomize the loss probabilities. Also, conditional on realization of P...

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In the Bernoulli mixture model, we randomize the loss probabilities. Also, conditional on realization of P = (P₁, ..., Pm) of the vector of random loss probabilities, the Bernoulli default indictator variable (L₁,..., Lm) are assumed to be independent. (a) Show that E[L₂] = E[P;], E[L;L;] = E[P.P;} = √" [" Pipj dF (Pi, Pj). Hint Use the property of conditional independence. (b) Show that the default correlation corr(L₁, L,) is given by cov (P₁, P₁) corr(Li, Lj): √E[P](1 - E[P])√/E[P;](1 – E[P;]) (c) Under the assumption of one-factor mixture model, where L₁ ~ B(1; p), with one single random default probability p. Show that corr(Li, Lj) in part (b) reduces to var(p) p(1-P)' where p = E(p). Under what condition that we can achieve corr(L₁, Lj) = 0? corr(L₁, Lj)= = [3] [3] [3] In the Bernoulli mixture model, we randomize the loss probabilities. Also, conditional on realization of P = (P₁, ..., Pm) of the vector of random loss probabilities, the Bernoulli default indictator variable (L₁,..., Lm) are assumed to be independent. (a) Show that E[L₂] = E[P;], E[L;L;] = E[P.P;} = √" [" Pipj dF (Pi, Pj). Hint Use the property of conditional independence. (b) Show that the default correlation corr(L₁, L,) is given by cov (P₁, P₁) corr(Li, Lj): √E[P](1 - E[P])√/E[P;](1 – E[P;]) (c) Under the assumption of one-factor mixture model, where L₁ ~ B(1; p), with one single random default probability p. Show that corr(Li, Lj) in part (b) reduces to var(p) p(1-P)' where p = E(p). Under what condition that we can achieve corr(L₁, Lj) = 0? corr(L₁, Lj)= = [3] [3] [3]