Let the six numbers a 1 , a 2 , . . . , a 6 be in A . P . and a 1 + a 3 = 10 .If the mean of these six numbers is 19 2 and their variance is σ 2 , then 8 σ 2 is equal to

Step-by-step solution

Let a 1 = a and common difference be d . Now, a 1 + a 3 = 10 ⇒ 2 a + 2 d = 10 ⇒ a + d = 5     . . . i Now, Given mean x ¯ = 19 2 ⇒ a + a + d + a + 2 d + . . . + a + 5 d 6 = 19 2 ⇒ 6 a + 15 d 6 = 19 2 ⇒ 2 a + 5 d = 19       . . . ii Solving i   &   ii , we get a = 2 ,   d = 3 So, A . P is 2 ,   5 ,   8 ,   11 ,   14 ,   17 . Now variance is given by, σ 2 = 1 6 ∑ i = 1 6 x i - x ¯ 2 ⇒ σ 2