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) 2.5 4 P(x = 4) = 4! (0.3) (1 0.3) = (5.5)(4.5)(3.5)(2.(0.0118) 4! = 0.1065. Step2: ii). To calculate what is the probability that at least one plant is found. P( at least one plant) = 1- P( zero plant) = 1- P(x = 0) 2.5 0 = 1 - k(2.5,0)(0.3) (1 0.3) = 1- 0.0493 ( Here, k(r , 0) = 1.) = 0.9507. Therefore, P(x = 4) = 0.1065 and the probability that at least one plant is 0.9507.