Let X be a random variable following binomial distribution with parameters n = 5 and p = k . Further, P(X = 1) = 0.4096 and P(X = 2) = 0.2048 . What is the value of k ?

Step-by-step solution

Given X follows a binomial distribution with parameters n = 5 and p = k . The probability mass function is given by P(X = x) = ^ n C_ x p^ x (1-p)^ n-x . For x = 1 : P(X = 1) = ⁵C₁ k¹ (1-k)⁴ = 5k(1-k)⁴ = 0.4096 For x = 2 : P(X = 2) = ⁵C₂ k² (1-k)³ = 10k²(1-k)³ = 0.2048 Dividing P(X = 1) by P(X = 2) , we get: P(X = 1) P(X = 2) = 5k(1-k)⁴ 10k²(1-k)³ 0.4096 0.2048 = 1-k 2k 2 = 1-k 2k 4k = 1 - k 5k = 1 k = 0.2