Free CAS MAS-I (Modern Actuarial Statistics I) Practice Questions

B is correct.

For a k-out-of-n system, a minimal path set consists of exactly k components. For the series combination of a 3-out-of-50 and 48-out-of-50 system, a minimal path set must contain a path set from each subsystem.

The 3-out-of-50 system has $\binom{50}{3} = 19{,}600$ minimal path sets. The 48-out-of-50 system has $\binom{50}{48} = \binom{50}{2} = 1{,}225$ minimal path sets.

For the series system, each minimal path set is the union of one from each. However, components may overlap. But since these are separate systems with separate components (100 components total), the number of minimal path sets is $19{,}600 \times 1{,}225 = 24{,}010{,}000$.

E is correct.

With constant force of mortality, the continuous whole life insurance APV is:
$$\bar{A}_x = \int_0^{\infty} e^{-\delta t} \mu e^{-\mu t}\,dt = \frac{\mu}{\mu + \delta} = \frac{0.04}{0.04 + 0.06} = \frac{0.04}{0.10} = 0.40$$

C is correct.

Let us evaluate each statement:

A: TRUE. Standardizing predictors before PCA is recommended because PCA is sensitive to the scale of variables.

D: FALSE. PCR does NOT perform feature selection in the traditional sense. It selects principal components (linear combinations of all original features), but it does not select or exclude individual features. All original variables contribute to each principal component. Feature selection methods like LASSO actually zero out coefficients.

C: TRUE. This is a known limitation/assumption of PCR -- it assumes that the directions of maximum variance in the predictors are the directions most associated with the response.

(B): TRUE. PCR can reduce overfitting by using fewer principal components than original features, effectively reducing dimensionality.

E: TRUE. By definition, the first principal component is the direction of maximum variance in the data.