Let the mean and variance of the observations x_n - 2n , for n=1, 2, , 5 , be 4 and 6 respectively. If x₁=3, x₂=8, x₃=a, x₄=13 and x₅=b with a b , then the variance of the observations x₁, x₂, x₃, x₄, x₅ is

Step-by-step solution

Let y_n = x_n - 2n . The observations y_n for n=1, 2, 3, 4, 5 are: y₁ = 3 - 2(1) = 1 y₂ = 8 - 2(2) = 4 y₃ = a - 2(3) = a - 6 y₄ = 13 - 2(4) = 5 y₅ = b - 2(5) = b - 10 Given, Mean of y_n = 4 1 + 4 + (a - 6) + 5 + (b - 10) 5 = 4 a + b - 6 = 20 a + b = 26 Given, Variance of y_n = 6 1^2 + 4^2 + (a - 6)^2 + 5^2 + (b - 10)^2 5 - 4^2 = 6 1 + 16 + (a - 6)^2 + 25 + (b - 10)^2 = 5 22 = 110 (a - 6)^2 + (b - 10)^2 = 68 Let u = a - 6 and v = b - 10 . Then u + v = 10 and u^2 + v^2 = 68 . (u + v)^2 = u^2 + v^2 + 2uv 100 = 68 + 2u