Let X = x ∈ N : 1 ≤ x ≤ 17 and Y = a x + b : x ∈ X and a , b ∈ R , a > 0 . If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to

Step-by-step solution

Y x ¯ = a x ¯ + b = a 1 + 2 + 3 + … . + 17 17 + b = 17 a 17 × 18 17 × 2 + b = 17 9 a + b = 17 σ X 2 = ∑ x 2 n - ∑ x n 2 = 1 2 + 2 2 + … + 17 2 17 - 1 + 2 + . . . + 17 17 2 = 17 · 18 · 35 6 · 17 - 17 · 18 2 · 17 2 = 105 - 81 = 24 ∴   σ Y 2 = a 2 σ X 2 = a 2 · 24 = 216 ⇒ a 2 = 216 24 = 9 a = 3 ∴   b = 17 - 27 ∴   b = 17 - 27 b = - 10 ∴   a + b = - 7