Final answer:
The expression E[ABCD] is simplified by considering t1, t2, t3, and t4 as deterministic variables that are constant, thus not affecting the probabilistic relationships between the variables A, B, C, and D. This relates to the expectation of the product of random variables and uses known probabilistic rules like the Multiplication Rule.
Step-by-step explanation:
To simplify the expression E[ABCD] = E[AB] * E[CD] + E[AC] * E[BD] + E[AD] * E[BC], we must realize that this is a problem related to the expectation of the product of random variables, which, in this case, can be solved by assuming that the variables t1, t2, t3, and t4 are deterministic and act like constants. When these deterministic variables represent specific outcomes or scalar components within a system, they can be inserted into any probabilistic equation without altering the overall behavior of other random variables involved. For instance, the Multiplication Rule in probability can be illustrated through the scenario where the probability of two events occurring together (A AND B) is the product of the probability of B and the conditional probability of A given B, shown as P(A AND B) = P(B)P(A|B).
However, when deterministic values are part of the expression, such as in physics or engineering equations, they simply represent known quantities that do not change or introduce randomness into the system. For example, when given x(t) = wt + at, if w and a are deterministic constants, solving for t is straightforward algebra. Thus, in our initial expression, by treating t1 through t4 as deterministic, we can simplify the relationship between E[ABCD] and its parts using known probabilistic identities and algebraic manipulation.