The generalized negative binomial pmf is given by Let X, the number of plants of a certain species found in a particular region, have this distribution with p = .3 and r =2.5. What is P( X = 4)? What is the probability that at least one plant is found?

Answer: Step1: Given, the negative binomial pmf is given by r x nb(x;r,p) = k(r,x). p (1 p) ; x = 0,1,2…. ------- (1) Let ‘x’ be the number of plants of a certain species found in a particular region. Given, p = 0.3 and r = 2.5. i). To find P(x = 4). k(r , x) = (x + r1)(x +r2)......(x +rx) x! Substitute this in equation (1) Therefore, (x + r1)(x +r2)......(x +rxr x nb(x;r,p) = x! p (1 p) (4+2.51)(4+2.52)(4+2.53)(4+2.54)...