Question: In the grafting context of Exercise, suppose that grafts are done one at a time and the process continues until the first failed graft. Let X denote the number of grafts up to and including the first failed graft.

(a) The random variable X is (choose one)

(i) binomial

(ii) hypergeometric

(iii) negative binomial

(iv) Poisson.

(b) Give the sample space and PMF of X.

(c) Give the expected value and variance of X.

Exercise: Grafting, the uniting of the stem of one plant with the stem or root of another, is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety with a hardy root system. For example, most sweet oranges grow on trees grafted to the root of a sour orange variety. Suppose each graft fails independently with probability 0.3. Five grafts are scheduled to be performed next week. Let X denote the number of grafts that will fail next week.

(a) The random variable X is (choose one)

(i) binomial

(ii) hypergeometric

(iii) negative binomial

(iv) Poisson.

(b) Give the sample space and PMF of X.

(c) Give the expected value and variance of X.

(d) Suppose that the cost of each failed graft is $9.00. Find:

(i) The probability that the cost from failed grafts will exceed $20.00.

(ii) The expected value and the variance of the cost from failed grafts.