Answer: t with it (imaginary unit).
- A t with -t, simply flipping the sign of the parameter
- B t with it (imaginary unit)
- C t with 1/t, inverting the parameter inside the MGF
- D t with t², squaring the parameter before substitution
Correct answer: B. t with it (imaginary unit)
Explanation: MGF: M(t) = E[e<sup>tX</sup>]. Characteristic function: φ(t) = E[e<sup>itX</sup>], where i = √(−1). This substitutes t → it. Unlike MGF, φ(t) always exists for all distributions since |e<sup>itX</sup>| = 1. Answer: t replaced by it.
Venn diagram of the universal set U with events A and B, showing the intersection (A and B), the parts unique to each event, and the complement region outside both.
Concept context
Chance, events, conditional probability, and Bayes theorem