Consider the following for the two (02) items that follow: Let X be a random variable following binomial distribution with parameters n = 6 and p = k . Further, 9P(X = 4) = P(X = 2) . What is the value of k ?

Step-by-step solution

The probability mass function of a binomial distribution is given by P(X = r) = ^ n C_ r p^r q^ n-r , where q = 1 - p . Given n = 6 , p = k , and q = 1 - k . We have 9P(X = 4) = P(X = 2) . Substituting the probabilities: 9 (⁶C₄) k^4 (1 - k)^2 = (⁶C₂) k^2 (1 - k)^4 Since ⁶C₄ = ⁶C₂ = 15 , we can cancel it from both sides: 9 k^4 (1 - k)^2 = k^2 (1 - k)^4 Assuming k 0 and k 1 , dividing both sides by k^2 (1 - k)^2 gives: 9 k^2 = (1 - k)^2 Taking the square root of both sides: 3k = 1 - k 4k = 1 k = 1 4 The other root 3k