Given the distribution with xᵢ = 1, 2, 3, ..., n and fᵢ = 1, 2⁻¹, 2⁻², ..., 2⁻⁽ⁿ⁻¹⁾. What is the mean of the distribution?

Mean = Σxᵢfᵢ / Σfᵢ. Σfᵢ = 1 + 1/2 + 1/4 + ... + 1/2ⁿ⁻¹ = (1 - (1/2)ⁿ)/(1 - 1/2) = 2(1 - 1/2ⁿ) = (2ⁿ⁺¹ - 2)/2ⁿ = (2ⁿ - 1)/2ⁿ⁻¹. From previous answer, Σxᵢfᵢ = (2ⁿ⁺¹ - n - 2)/2ⁿ⁻¹. Therefore Mean = [(2ⁿ⁺¹ - n - 2)/2ⁿ⁻¹] / [(2ⁿ - 1)/2ⁿ⁻¹] = (2ⁿ⁺¹ - n - 2)/(2ⁿ - 1).