Question

Consider that a sample of size 3 is randomly drawn from a population that takes only two values, equally likely: -1 and 1. Let \( z = \max(x_1, x_2, x_3) \) where \( x_1, x_2, x_3 \) are the sample observations. The expected value of \( z \), \( E(z) \) is _________ (round off to two decimal places).

Hint:

When calculating expected value for discrete random variables, multiply each value by its probability and sum the results.

Answer 1

Correct Answer: 0.7

The possible values for \( x_1, x_2, x_3 \) are -1 and 1, and \( z = \max(x_1, x_2, x_3) \). Since the values of \( x_1, x_2, x_3 \) are equally likely, there are 8 possible combinations. The probability that \( z = 1 \) (the maximum of the three values is 1) is the complement of the probability that \( z = -1 \). For \( z = -1 \), all three values must be -1, which occurs with probability \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \). Thus, the probability that \( z = 1 \) is \( 1 - \frac{1}{8} = \frac{7}{8} \). The expected value of \( z \) is: \[ E(z) = 1 \times \frac{7}{8} + (-1) \times \frac{1}{8} = \frac{7}{8} - \frac{1}{8} = \frac{6}{8} = 0.75 \] Thus, the expected value of \( z \) is \( 0.75 \).