Discussion
The cumulative distribution function F of a random variable X is given by :
\[ \begin{aligned} F(x) &= 0.2\,\Delta_{0}(x) + 0.4\,\Delta_{1}(x) \\ &\quad + 0.2\,\Delta_{2}(x) + 0.2\,\Delta_{3}(x), \end{aligned} \]
where \(\Delta_{a}\) is the degenerate distribution function degenerate at 'a'. The following are two statements :
(I) \(P(X = 1 \mid X \le 1.5) = 2/3\)
(II) Variance (X) = 0
Which of the following is true ?
Answer: B
Only (I) is true.
The distribution function puts mass 0.2 at 0, 0.4 at 1, 0.2 at 2 and 0.2 at 3. For (I):
\[ \begin{aligned} P(X = 1 \mid X \le 1.5) &= \frac{P(X = 1)}{P(X \in \{0, 1\})} \\ &= \frac{0.4}{0.6} = \frac{2}{3} \end{aligned} \]
so (I) holds.
For (II), a variance of 0 would mean X is a single constant, but X takes four values. Indeed \(E(X) = 0.4 + 0.4 + 0.6 = 1.4\), \(E(X^{2}) = 0.4 + 0.8 + 1.8 = 3\), so \(\text{Var}(X) = 3 - 1.96 = 1.04\). Statement (II) confuses the components, each of which is degenerate, with the mixture.
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