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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 withprobability 0.3. Five grafts are scheduled to be performed nextweek. Let X deonte the number of graftes that will fail nextweek.
a. The random variable x is (choose one): binomial, hypergeometric, negative binomial, 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 and the variance of the cost from failed grafts.

Sagot :

ogorwyne

Step-by-step explanation:

a. The random variable is a binomial distribution.

b. the sample space, X = {0, 1, 2, 3, 4, 5}

the pmf

we solve for this using

nCx * P^x * (1-p)^n-x

n = 5

p = 0.3

for x = 0

5C0 * 0.3⁰(1-0.3)^5-0

= 1 * 1* 0.7⁵

= 0.16807

for x = 1

5C1*0.3¹(1-0.3)^5-1

= 5*0.3(0.7)⁴

= 5x0.3x0.2401

= 0.36015

for x = 2

5C2 * 0.3² * (1-0.3) ^5-2

= 0.30870

for x = 3

5C3 * 0.3³ * (1-0.3) ^ 5-3

= 10 * 0.027 * 0.7²

= 0.1323

for x = 4

5C4 * 0.3⁴ (1-0.3) ^ 5-4

= 5 * 0.0081 * 0.7

= 0.02835

for x = 5

5C5 *0.3⁵ (1- 0.3) ^5-5

= 1*0.00243*0,7⁰

= 0.00243

c. E[X] = N*p = 5*0.3 = 1.5

var[X] = np(1-p) = 5*0.3*0.7 = 1.05

d. 20/9 = 2.222

so we have that if x is greater than or equal to 3, cost will exceed 20

p(x=3) + p(x=4) + p(x=5)

= 0.1323 + 0.02835 + 0.00243

probability = 0.16308

E[C] = 1.5 * 9 = 13.5

VAR[C] = 1.05 * 9 = 9.45

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