Discussion
Let \(F(x) = P[X \le x]\) be the distribution function of a random variable X.
Define \(D_{n} = \left\{u \mid F(u) - F(u-) \ge \frac{1}{n}\right\}\), \(n = 1, 2, 3, \ldots\)
Then :
Answer: A
The true statement is that \(D_{n}\) cannot contain more than n elements.
\(F(u) - F(u-)\) is the jump of F at u, which equals \(P(X = u)\). Every point of \(D_{n}\) carries probability at least \(\frac{1}{n}\), and the total probability is 1, so there can be at most n such points.
The other statements fail. For a continuous distribution (normal, say) F has no jumps, so every \(D_{n}\) is empty. \(D_{2}\) has 2 elements only when there are two atoms of mass at least \(\frac{1}{2}\) each, as in a fair coin, and not 'always'. \(D_{1}\) is not always empty: for a degenerate distribution, the single atom has mass 1.
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