Final Answer:
a) The event (N-2) in terms of A, B, and C is the intersection of exactly (N-2) events not occurring, which can be expressed as (1-A)(1-B)(1-C) raised to the power of (N-2).
b) The probability of event (N-2) in terms of a, b, and c is (1-a)(1-b)(1-c) raised to the power of (N-2).
c) The expected value of N in terms of a, b, and c is E(N) = a + b + c.
d) The standard deviation of N in terms of a, b, and c is SD(N) = sqrt(a(1-a) + b(1-b) + c(1-c)).
Step-by-step explanation:
For event (N-2), since N is the random number of events occurring, the complement of an event occurring is (1 - probability of the event occurring). Therefore, the event (N-2) represents the intersection of exactly (N-2) events not occurring. In terms of A, B, and C, this is denoted by (1-A)(1-B)(1-C) raised to the power of (N-2), as each individual event not occurring is the complement of its probability.
To find the probability of event (N-2) in terms of a, b, and c, we substitute their complements into the expression, resulting in (1-a)(1-b)(1-c) raised to the power of (N-2). This reflects the probability that exactly (N-2) events out of the mutually independent events A, B, and C do not occur.
The expected value (E(N)) of N, given mutually independent events, is the sum of their individual probabilities. Hence, E(N) = a + b + c represents the total probability of occurrences across the events A, B, and C.
Regarding the standard deviation (SD(N)), for mutually independent events, the variance of each event is given by p(1-p), where p is the probability of an event occurring. By summing the variances of A, B, and C, and taking the square root, SD(N) = sqrt(a(1-a) + b(1-b) + c(1-c)) represents the standard deviation of N in terms of a, b, and c. This reflects the measure of the dispersion or spread of the random variable N around its expected value.