问题及解答

Suppose $E(X)=5$ and $E[X(X-1)]=27.5$. What is

• $E(X^2)$ ? [Hint: $E[X(X-1)]=E[X^2-X]=E(X^2)-E(X)$.]
• $V(X)$ ?
• The general relationship among the quantities $E(X)$, $E[X(X-1)]$, and $V(X)$.

Solution.

(a)

\[
E[X(X-1)]=E[X^2-X]=E(X^2)-E(X)
\]

Hence,

\[
E(X^2)=E[X(X-1)]+E(X)=27.5+5=32.5
\]

(b)

\[
V(X)=E(X^2)-[E(X)]^2=32.5-5^2=7.5
\]

(c)

By the formula

\[
\begin{aligned}
V(X)&=E(X^2)-[E(X)]^2\\
E(X^2)&=E[X(X-1)]+E(X)
\end{aligned}
\]

we infer that

\[
V(X)=E[X(X-1)]+E(X)-[E(X)]^2
\]