Use stated constructions
The first two questions use five independent identically distributed normal period losses with mean 2 and dispersion 10. The aggregate is an arithmetic sum. The final question distinguishes separate normal marginals from a joint normal law.
Retain the supplied assumptions
No empirical independence or official syllabus coverage is implied. Explain which model condition justifies a formula before applying its arithmetic.
Try the questions
What is the mean of the five-period independent sum?
- A.
10
- B.
2√5
- C.
2
Answer and explanation
Answer: A
The mean scales with the number of additive periods.
- A
Means add: 5×2.
- B
This incorrectly scales the mean like dispersion.
- C
This omits the other periods.
What is the sum’s standard deviation under the stated independence?
- A.
10√5
- B.
50
- C.
500
Answer and explanation
Answer: A
The aggregate variance is 5×100.
- A
Independent variances add before taking the square root.
- B
This fits the separately stated perfect shared-shock construction.
- C
This mixes variance and standard deviation units.
Each component marginal is normal. Does that alone establish a normal sum?
- A.
Yes, regardless of dependence
- B.
No; the joint law still matters
- C.
Yes if covariance is zero
Answer and explanation
Answer: B
X and SX provide a counterexample with normal marginals and zero covariance.
- A
The original sign construction creates a nonnormal sum.
- B
Joint normality is an additional sufficient condition.
- C
Zero covariance does not generally establish independence or joint normality.