Question-75

Consider \(n\) bits \(X_1,\ldots,X_n\), where each bit is equally likely to be \(0\) or \(1\), and is independent of all other bits. Define \(n-1\) bits \(Y_i=X_iX_{i+1}\), \(i=1,\ldots,n-1\). Let \(N_X\) and \(N_Y\) be, respectively, the number of 1s in \(\{X_1,\ldots,X_n\}\) and \(\{Y_1,\ldots,Y_{n-1}\}\). Assuming \(n = 100\), what is the expected value of \(N_X\) and \(N_Y?\)

• \(E(N_X) = 50, E(N_Y) = 49.5\)

• \(E(N_X) = 50, E(N_Y) = 24.75\)

• \(E(N_X) = 100, E(N_Y) = 50\)

• \(E(N_X) = 100, E(N_Y) = 99\)

Hint

Answer

• \(E(N_X) = 50, E(N_Y) = 49.5\)

• \(E(N_X) = 50, E(N_Y) = 24.75\)

• \(E(N_X) = 100, E(N_Y) = 50\)

• \(E(N_X) = 100, E(N_Y) = 99\)

Solution