Consider n random variables X1, X2, ..., Xn that can take the values 0 and 1. Here, n is an integer greater than or equal to 4. The probability of an event A is denoted by P(A), and the conditional probability of the event A given an event B is denoted by P(A|B). The intersection between the event A and the event B is denoted by A ∧ B. Answer the following questions.
I. Let us assume that the X1, X2, ..., Xn are independent. In addition, assume that each Xk (k = 1, 2, ..., n) takes the value 1 with the probability p and the value 0 with the probability 1 - p, i.e., P(Xk = 1) = p and P(Xk = 0) = 1 - p.
1. Find the expected value and the variance of the sum of the X1, X2, ..., Xn.
2. The random variables X1, X2, ..., Xn are arranged in the row Xn ··· X2X1. Let Y be the integer value obtained by regarding that row as an n-digit binary number. For example, in the case that n = 4, Y = 5 when the row X4X3X2X1 is 0101, and Y = 13 when the row X4X3X2X1 is 1101. Y is a random variable that takes integer values from 0 to 2^n - 1. Obtain the expected value and variance of Y.