Answer: E[etX].
- A E[X]
- B E[X²]
- C E[etX]
- D E[tX]
Correct answer: C. E[etX]
Explanation: MGF: M(t) = E[etX]. Expanding etX = 1 + tX + t²X²/2! + … gives M(t) = 1 + tE[X] + t²E[X²]/2! + …. The n-th derivative M⁽ⁿ⁾(0) = E[Xⁿ], the n-th raw moment. Answer: E[etX].
In a perfectly normal (bell-shaped) distribution, mean, median, and mode all coincide at the centre; about 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 - the empirical rule used to judge how typical or extreme a value is.
Concept context
Mean, median, mode, standard deviation, and data interpretation