Solutions - 5240-05

(1) Define indicator variables $X_A$ and $X_B$:

Let $X_A$ denote whether $A$ occurs and $X_B$ denote whether $B$ occurs. They are independent since $A$ and $B$ are independent ($P[AB]=P[A]P[B]$). Let $X=X_A+X_B$.

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(2) Compute $\text{Var}[X]$:

$$\begin{array}{cc}\text{Var}[X]& =\text{Var}[X_A+X_B]\\ & =\text{Var}[X_A]+\text{Var}[X_B]\\ & =(E\left[X_A^2\right]-{\left(E[X_A]\right)}^2)+(E\left[X_B^2\right]-{\left(E[X_B]\right)}^2)\\ & =(0.5-{0.5}^2)+(0.2-{0.2}^2)\\ & =0.25+0.16=0.41\end{array}$$