Fair And Biased Coins
Say I flip 200 coins, 100 of which are fair, and 100 of which land on heads with probability \(\frac{1}{4}\).
What is the expected number of heads?
Let \(R_i\) be an indicator variable for the \(i^{th}\) fair coin, where \(R_i = \begin{cases} 1 & i^{th}\text{ flip is heads} \\ 0 & i^{th}\text{ flip is tails} \end{cases}\).
Similarly, define \(S_j\) as the indicator variable for the \(j^{th}\) biased coin.
Then, the number of heads is given by \(H=R_1+R_2+\ldots +R_{100}+S_1+S_2+\ldots+S_{100}\).
By linearity of expectation, \[E[H]=E[R_1+R_2+\ldots +R_{100}+S_1+S_2+\ldots+S_{100}]\] \[=E[R_1]+E[R_2]+\ldots +E[R_{100}]+E[S_1]+E[S_2]+\ldots+E[S_{100}]\] \[=100\cdot\frac{1}{2}+100\cdot\frac{1}{4}=50+25=75.\]
Similarly, define \(S_j\) as the indicator variable for the \(j^{th}\) biased coin.
Then, the number of heads is given by \(H=R_1+R_2+\ldots +R_{100}+S_1+S_2+\ldots+S_{100}\).
By linearity of expectation, \[E[H]=E[R_1+R_2+\ldots +R_{100}+S_1+S_2+\ldots+S_{100}]\] \[=E[R_1]+E[R_2]+\ldots +E[R_{100}]+E[S_1]+E[S_2]+\ldots+E[S_{100}]\] \[=100\cdot\frac{1}{2}+100\cdot\frac{1}{4}=50+25=75.\]