If X is a Poisson variate such that 2 P ( X = 1 ) = 5 P ( X = 5 ) + 2 P ( X = 3 ) , then the standard deviation of X is

Step-by-step solution

It is given that, 2 P ( X = 1 ) = 5 P ( X = 5 ) + 2 P ( X = 3 ) , The formula for Poisson P X = r = e - λ λ r r ! . ⇒ 2 e - λ λ 1 ! = 5 e - λ λ 5 5 ! + 2 e - λ λ 3 3 ! ⇒ 2 = λ 4 24 + λ 2 3 ⇒ 2 = 3 λ 4 + 24 λ 2 72 ⇒ 144 = 3 λ 4 + 24 λ 2 ⇒ λ 4 + 8 λ 2 - 48 = 0 Consider λ 2 = x , ⇒ x 2 + 8 x - 48 = 0 ⇒ ( x + 12 ) ( x - 4 ) = 0 x = 4 and x ≠ - 12 The standard deviation is, x = λ 2 =