For a binomial variate X with parameters n = 5 and p = 3 4 . If α = 1 9 P ( X ≥ 3 ) and β = P ( X ≤ 2 ) , then 256 ( β - α ) =

Step-by-step solution

We know that the binomial distribution is given by P X = r = C r n p r q n - r and p + q = 1 . Now, we have n = 5 ,   p = 3 4 ,   q = 1 4 Now, α = 1 9 P ( X ≥ 3 ) ⇒ α = 1 9 P ( X = 3 ) + P ( X = 4 ) + P ( X = 5 ) = 1 9 C 3 5 3 4 3 1 4 2 + C 4 5 3 4 4 1 4 1 + C 5 5 3 4 5 1 4 0 = 102 4 5 And, β = P ( X ≤ 2 ) = P ( X = 0 ) + P ( X = 1 ) + P ( X = 2 ) = C 0 5 3 4 0 1 4 5 + C 1 5 3 4 1 1 4 4 + C 2 5 3 4 2 1 4 3 = 106 4 5 Therefore, 256 β - α = 256 106 4 5 - 102 4 5 =