Discussion
Suppose (X, Y) is a two-dimensional random vector with range \((0, \infty) \times \{1, 2\}\), such that for any \(A \subset (0, \infty)\) and \(y = 1, 2\),
\[ P(X \in A, Y = y) = \frac{1}{2}\int_{A} ye^{-yx}\,dx. \]
Then, which of the following statements is false ?
Answer: B
The false statement is B: the marginal distribution function of X is not \(1 - \tfrac{1}{2}[e^{-x} + 2e^{-2x}]\).
Summing the joint law over \(y = 1, 2\) gives the marginal density of X:
\[ f_{X}(x) = \tfrac{1}{2}[e^{-x} + 2e^{-2x}],\ x \gt 0 \]
which confirms C. Integrating from 0 to \(x\):
\[ \begin{aligned} F(x) &= \tfrac{1}{2}(1 - e^{-x}) + \tfrac{1}{2}(1 - e^{-2x}) \\ &= 1 - \tfrac{1}{2}\left[e^{-x} + e^{-2x}\right] \end{aligned} \]
Option B keeps the factor 2 from the density, which disappears on integration; it even gives \(F(0) = -1/2\). D is true because \(P(Y = y) = \tfrac{1}{2}\int_{0}^{\infty} ye^{-yx}dx = \tfrac{1}{2}\). A is true: Y is discrete and X continuous, so (X, Y) has neither a joint mass function nor a joint density on the plane.
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