Free SOA Exam P (Probability) Joint & Marginal Distributions Practice Questions

$\text{Cov}(X, 3X+5)$ = 3$\cdot \text{Var}(X)$ + 0 = 3(4) = 12.

$P(X+Y < 1)$ = area of triangle below x+y=1 in unit square = $\frac{(1)}{2}$.

$$\text{Cov}(X, Y) = E[XY] - E[X] \cdot E[Y] = E[X^3] - E[X] \cdot E[X^2]$$

For $X \sim N(0,1)$, all odd moments are zero by symmetry: $E[X] = 0$, $E[X^3] = 0$. Also $E[X^2] = 1$.

$$\text{Cov}(X, X^2) = 0 - 0 \cdot 1 = 0$$

$\text{Cov}(X, Y) = 0$ does NOT imply independence. Here $Y = X^2$ is a deterministic function of $X$, so $X$ and $Y$ are completely dependent despite being uncorrelated. covariance only captures linear relationships, and the relationship $Y = X^2$ is purely nonlinear.

Distractor analysis:
- (A) $1$: assumes $E[X^3] = E[X^2] = 1$ or uses $\text{Cov}(X, X^2) = \text{Var}(X)$.
- (B) $2$: computes $\text{Var}(X^2) = E[X^4] - (E[X^2])^2 = 3 - 1 = 2$ and confuses it with covariance.
- (D) $-1$: assumes a negative linear association since $X$ can be negative while $X^2$ is positive.
- (E) $\frac{1}{2}$: applies a nonexistent formula $\text{Cov}(X, X^2) = \frac{1}{2}E[X^2]$.

The answer is $0$.