独立事件 (Indenpent Event)

(1) $$\begin{array}{ll}P\left( {A \cap B’} \right) &= P\left( A \right) – P\left( {A \cap B} \right) = P\left( A \right) – P\left( A \right)P\left( B \right) \\&= P\left( A \right) \cdot \left( {1 – P\left( B \right)} \right) = P\left( A \right)P\left( {B’} \right)\end{array}$$，得证。参阅图一。

(2) 的证明与(1)相同。

(3) $$\begin{array}{ll}P\left( {A’ \cap B’} \right) &= 1 – P\left( {A \cup B} \right) = 1 – P\left( A \right) – P\left( B \right) + P\left( {A \cap B} \right) \\&= 1 – P\left( A \right) – P\left( B \right) + P\left( A \right)P\left( B \right) \\&= \left( {1 – P\left( A \right)} \right) \cdot \left( {1 – P\left( B \right)} \right) = P\left( {A’} \right) \cdot P\left( {B’} \right)\end{array}$$

$$\displaystyle\begin{array}{ll}E\left( {XY} \right) &= \sum {{x_i}{y_i}{p_i}}\\&=\frac{1}{{36}}(1 + 4 + 6 + 12 + 10 + 24 + 16 + 9 + 20 + 48 + 30 + 16 + 36 + 40 + 48 + 25 + 60 + 36)\\&=\frac{1}{36}\times 441 = \frac{49}{4}\end{array}$$

$$\displaystyle\begin{array}{ll}E\left( X \right)E(Y) &= (\sum\limits_{i = 1}^6 {{x_i}{p_i}} )(\sum\limits_{i = 1}^6 {{y_i}{p_i}} )\\&=\left( {1 \times \frac{1}{6} + 2 \times \frac{1}{6} + 3 \times \frac{1}{6} + 4 \times \frac{1}{6} + 5 \times \frac{1}{6} + 6 \times \frac{1}{6}} \right) \cdot \left( {1 \times \frac{1}{6} + 2 \times \frac{1}{6} + 3 \times \frac{1}{6} + 4 \times \frac{1}{6} + 5 \times \frac{1}{6} + 6 \times \frac{1}{6}} \right)\\&= \frac{{1 + 2 + 3 + 4 + 5 + 6}}{6} \times \frac{{1 + 2 + 3 + 4 + 5 + 6}}{6}\\&=\frac{7}{2}\times\frac{7}{2}=\frac{49}{4}\end{array}$$

$$P\left( A \right) = {\left( {\frac{1}{2}} \right)^2} + C_1^2 \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{4} = \frac{{12}}{{16}}$$

$$P\left( B \right) = {\left( {\frac{1}{2}} \right)^4} + C_3^4{\left( {\frac{1}{2}} \right)^3} \cdot \frac{1}{2} + C_2^4{\left( {\frac{1}{2}} \right)^2} \cdot {\left( {\frac{1}{2}} \right)^2} = \frac{{11}}{{16}}$$，所以，$$P(A)>P(B)$$。

Papoulis, A. (1984), Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill, p. pp. 139-152 .http://en.wikipedia.org/wiki/Independence_(probability_theory).