If a variable takes values   0 ,   1 ,   2 , . . . , n with frequencies   2 n   ,   C 1   n · p q n - 1 ,   n C 2 p 2 · q n - 1 , … … … ,   n C n p n respectively, then the mean of the distribution is (where p + q = 1 ).

Step-by-step solution

∴ x ¯ = 0 · q n + 1 · n 1 q n - 1 p + 2 · n n - 1 2 ! q n - 2 p 2 + ... + n · p n q n + n 1 q n - 1 p + n n - 1 2 q n - 2 p 2 + . .. + p n = 0 · n C 0 q n p 0 + 1 · n C 1 q n - 1 p + . .. + n n C n q 0 p n n C 0 q n p 0 + n C 1 q n - 1 p 1 + . .. + n C n q n - n p n = Σ r = 0 n r n · C r q n - r p r Σ r = 0 n n C r q n - r p r = Σ r = 0 n r · n r n - 1 C r - 1 q n - r p · p r - 1 Σ r = 0 n n C r q n - r p r = n p p + q n - 1 q + p n = n p &