Discussion
Suppose that X is a random variable with c.d.f. F, where \(F(x) = 0.7\,\Delta_{1}(x) + 0.3\,N_{(0,\,1)}(x)\), where \(\Delta_{1}\) is the degenerate distribution function degenerate at 1 and \(N_{(0,\,1)}\) is the standard normal distribution function. Then, which of the following statements is true ?
Answer: C
E(X) = 0.7 and variance (X) = 0.51.
X is a mixture: with probability 0.7 it equals 1, and with probability 0.3 it is standard normal.
\[ \begin{aligned} E(X) &= 0.7(1) + 0.3(0) = 0.7 \\ E(X^{2}) &= 0.7(1) + 0.3(1) = 1 \\ \mathrm{Var}(X) &= 1 - 0.7^{2} = 0.51 \end{aligned} \]
The normal component contributes \(E(Z^{2}) = 1\), its variance plus its zero mean squared. A variance of 1 forgets to subtract \(0.49\), and 0.30 is only the weighted variance of the normal part, ignoring the spread between the two components.
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