Business failures are due to three mutually exclusive risks: market risk, credit risk, and operation risk, which account for 20%, 30%, and 50%, respectively, of all business failures. Suppose the number of business failures each year is Poisson distributed with mean 4.6.
(a) What is the chance that there are two business failures due to operation risk in a year?
(b) What is the chance that the business failures due to market risk and credit risk are both fewer than two in a year?
(c) Given that there are four business failures in a year, what is the probability that two of these are due to market risk?
could you kindly remove this question from "on-hold"? My comments are below.
a) What I was initially thinking, was that since there are 4.6 business failures a year on average (lambda = 4.6), then there are on average 0.5*4.6 = 2.3 operation risks a year right? so for a), do i use poisson with lambda = 2.3 to compute P(X=2)? – Suraj 7 hours ago
similarly for b), do i use poisson (2.3) (because its (0.2+0.3)*4.6 = 2.3 to find out the biz failures by market and credit risk? Then compute P (X<2)
(a) You are correct. Use Poisson with mean 2.3 and find $P(X=2)=\dfrac{(2.3^2)*e^{-2.3}}{2!}= 0.2652$
(b) There are two Poisson processes with means (4.6)$*$(0.2)=0.92 and (4.6)(0.3)=1.38. So, the answer should be $P(X<2 and Y<2)=P(X<2)*P(Y<2)=(P(X=0)+P(X=1))*(P(Y=0)+P(Y=1))=0.4582$
(c) It is a Binomial random variable with BN(4,0.2). Thus, $P(X=2)=\dbinom{4}{2}*0.2^2*0.8^2=0.1535$.