If a variable takes the values as 0, 1, 2,....n with frequencies as n C 0 , n C 1 , n C 2 , ....., n C n respectively, then mean and variance are given as :

Step-by-step solution

We know that x ¯ = ∑ f ⁡ x ∑ f ⁡ and σ = ∑ f ⁡ x 2 ∑ f ⁡ - x ¯ 2 Here Σ f = 1 + n C 1 + n C 2 + ... +n. n C n = 2 n Σ f x = 0.1 + 1 . n C 1 + 2. n C 2 +3. n C 3 +...+n. n C n = n.2 n-1 ​ Σfx 2 = 0.1 + 1 2 . n C 1 + 2 2 . n C 2 + 3 2 . n C 3 + ... + n 2 . n C n = n 2 n-1 + n (n-1) 2 n-2 TO obtain the above results, let 1 + y 2 = 1 + n C 1 y + n C 2 y 2 + … … + n C n y n … i put y = 1, 2 n = 1 + n C 1 + n C 2 + ... + n C n = Σf Differentiati