Let a random variable X has a binomial distribution with mean 8 and variance 4. If P X ≤ 2 = k 2 16 , then the value of k is equal to

Step-by-step solution

Let n be the number of times the experiment is done, p is the probability of the success and q is the probability of the failure. ∴ n p = 8 (mean), n p q = 4 (variance) ⇒ q = 1 2 ⇒ p = 1 2 ⇒ n = 16 ⇒ P X ≤ 2 = p X = 0 + p X = 1 + p X = 2 ⇒ P X ≤ 2 = 16 C 0 p 0 q 16 + 16 C 1 p 1 q 15 + 16 C 2 p 2 q 14 ⇒ P X ≤ 2 = 1 2 16 + 16 1 2 16 + 120 × 1 2 16 ⇒ P X ≤ 2 = 1 2 16 137 = 137 2 16 = k 2 16 ∴   k = 137